1 Libraries and environment

1.1 Load environment

Libraries used to create and generate this report:

R : R version 4.3.1 (2023-06-16)
rmarkdown : 2.21
knitr : 1.42
rmdformats : 1.0.4
bookdown : 0.34
kableExtra : 1.3.4

1.2 Load libraries

Libraries used to analyse data:

library("SNFtool")
library("pheatmap")
library("igraph")

SNFtool: 2.3.1
pheatmap: 1.0.12
igraph: 1.4.2

Libraries used to load data:

library("MOFAdata")
library("data.table")
library("mixOmics")

MOFAdata: 1.16.1
data.table: 1.14.8
mixOmics: 6.24.0

1.3 Visualization using Cytoscape

Cytoscape is used for visualization. Figures were generated using the Cytoscape v3.9.1 and several Cytoscape apps:

yFiles Layout Algorithm: 1.1.3
LegendCreator: 1.1.6

2 General principle of the SNF method

Similarity Network Fusion (SNF) builds networks of samples for each data type. Then, it fuses them into one network, which represents the full spectrum of underlying data.

In other words, SNF integrates several types of data (e.g. omics data) into one network which represents the relationships between samples.

The SNF methods can be decomposed into three main steps, displayed in the Figure 2.1 (note that patient=samples in the description below):

First, for each data type given as input (a), it calculates the patient similarity matrix (b)
Then, from each similarity matrix (b), it creates the patient similarity network (c)
Finally, it fuses the different patient similarity networks (d) into one fused similarity network (e)

Figure 2.1: Similarity Network Fusion method overview. The figure is comming from Wang et al., 2014.

In the final fused similarity network (e), you can identify which data type contributes to which edge:

the blue edge information are supported by the mRNA data
the pink edge information are supported by the methylation data
the orange edge information are supported by both data type: mRNA and methylation.

You can retreive more information in:

GitHub page: https://github.com/maxconway/SNFtool
Publication: PMID: 24464287

3 Choose your datasets

Choose the dataset on which you want to apply SNF!!

Different datasets are available. Note that each dataset has its specificity and some analysis steps should be adapted.

Four datasets are available: **Metagenomic** dataset from Tara Ocean (image from Sunagawa et al., 2015), **Breast cancer** dataset from TCGA (image from TCGA [website](https://portal.gdc.cancer.gov/)), **CLL** dataset (Dietrich et al., 2018) and **tomato plant** dataset (figure from google image).

Figure 3.1: Four datasets are available: Metagenomic dataset from Tara Ocean (image from Sunagawa et al., 2015), Breast cancer dataset from TCGA (image from TCGA website), CLL dataset (Dietrich et al., 2018) and tomato plant dataset (figure from google image).

3.1 Metagenomic dataset from Tara Ocean project

To retrieve data: download files from summer school’s GitHub repository

dataset:
- TARAoceans_proNOGS.cvs
- TARAoceans_proPhylo.csv
metadata:
- TARAoceans_metadata.csv

Samples come from eight oceans around the world (SPO: South Pacific Ocean, NAO: North Atlantic Ocean, IO: Indian Ocean, RS: Red Sea, MS: Mediterranean Sea, NPO: North Pacific Ocean, SO: Southern Ocean, SAO: South Atlantic Ocean).

Samples can come from different layers with different temperatures:

SRF: Surface Water Layer (0-5 meters)
DCM: Deep Chlorophyll Maximum (peak of chlorophyll, 0-600 meters)
MIX: Subsurface epipelagic Mixed Layer
MES(O): Mesopelagic zone (from 500/1000 meters)

In a previous analysis (Sunagawa et al., 2015), they identified a stratification mostly driven by the temperature rather than geography or other environmental factors.

We have two types of data:

orthologous genes: the relative abundance of groups of orthologous genes (OGs)
phylogenetic profil: counts of S16 rRNA

Does an integrative analysis of these two data types retrieve the stratification driver by the layers? Does it also find a geographical clustering?

Data are coming from: MiBiOmics gitlab.

3.2 Breast cancer dataset from The Cancer Genome Atlas

To retrieve data:

dataset: using data("breast.TCGA") from the mixOmics R package
- breast.TCGA$data.train$mirna
- breast.TCGA$data.train$mrna
- breast.TCGA$data.train$protein
metadata: using data("breast.TCGA") from the mixOmics R package
- breast.TCGA$data.train$subtype

Human breast cancer is a heterogeneous disease. Breast tumors can be classified into several subtypes (PAM50 classification), according to the mRNA expression level (Sorlie et al., 2001). In this dataset, we have three subtypes:

Basal: considered more aggressive than LumA
Her2: tend to grow faster than LumA and can have a worse prognosis, but are usually successfully treated
LumA: tend to grow more slowly than other cancers, be lower grade, and have a good prognosis

We have three types of data:

mRNA: mRNA expression level
miRNA: microRNA expression level
protein: protein abundance

Does an integrative analysis of these three data types retrieve the classification of the breast cancer? Or find another classification?

Data are coming from the mixOmics R package. The full data can be downloaded here.

3.3 Chronic Lymphocytic Leukaemia (CLL) dataset

To retrieve data:

dataset: using data("CLL_data") from the MOFAdata R package
- CLL_data_t$Drugs
- CLL_data_t$Methylation
- CLL_data_t$mRNA
- CLL_data_t$Mutations
metadata: download file from summer school’s GitHub repository
- sample_metadata.txt

The Chronic Lymphocytic Leukaemia (CLL) is type of blood and bone marrow cancer. The full data are explained in Dietrich et al., 2018 and available here.

We have four types of data:

mRNA: transcriptom expression level
methylation: DNA methylation assays
drug: drug response measurements
mutation: sommatic mutation status

3.4 Tomato plant dataset

To retrieve data: download files from summer school’s GitHub repository

dataset:
- mrna.tsv
- prots.tsv
metadata:
- samples_metadata.tsv

In order to study the protein turnover in developing tomato fruit (Solanum lycopersicum) in Belouah et al., two omics data types were collected:

transcript data: gene abundance
protein data: protein abundance

Each data type was collected in nine different developmental stages: GR1, GR2, GR3, GR4, GR5, GR6, GR7, GR8 and GR9. For each developmental stages, we have three replicates.

Does an integrative analysis of these data types retrieve the different developmental stages?

Data are coming from Belouah et al., 2019.

4 Input data

4.1 In summary

The preprocessing step is the most important part of the analysis. Data need to be prepared correctly in order to extract relevant information and produce a correct and pertinent interpretation of the results.

Data preprocessing could be summarized by four main steps:

Prepare the data (e.g. remove outliers, correct bacth effect etc…)
Remove and/or impute missing data
- remove features/samples if more than 20% of missing data
- impute missing data (e.g. using K-nearest neighbor method (KNN))
Normalize the data according to the data type
Scale (mean = 0 and standard deviation = 1)

Distribution examples expected after preprocessing

Figure 4.1: Distribution examples expected after preprocessing

The data must conform to a specific matrix shape:

samples (e.g. samples, organisms …) in rows
features (e.g. genes, proteins …) in columns

4.2 Read and prepare the data

HELP!

To access the documentation for a function within R, you can use the command ?functionName().
Take advantage of this command, it will be your best friend ! ;)

For this tutorial, we assume that the data have been already prepared: outliers are already removed and there is no batch effect.

4.2.1 Load dataset

4.2.1.1 Load from file

To load data from a file, you can use read.table() and specify the file name, the sep character and others parameters if it’s necessary.

4.2.1.2 Load from package

To load data from a package, you can use data(dataName). Don’t forget to load the corresponding package before with library(packageName).

4.2.1.3 Load from website

To load data from a website, you can use fread(url) from the data.table package.

4.2.2 Load metadata

The metadata contains complementary information about samples. You can load the metadata from package or file. See the section 4.2.1 about data loading.

We use mainly the metadata for visualization. So we suggest to follow these recommendations:

metadata should be a data frame (use data.frame() or as.data.Frame() functions)
row names of the data frame should be the sample names (from read.table() use the row.names = 1 parameter)
the data frame needs to contain only character or numerical (check using str() function)
select the columns that are the most useful to describe/characterize your samples

4.2.3 Practice

For instance, to load the ortologous gene data from Tara Ocean, the command could be:

tara_nog <- read.table(file = "../00_Data/TaraOcean_mibiomics/TARAoceans_proNOGS.csv", sep = ",", head = TRUE, row.names = 1)
tara_nog[c(1:5), c(1:5)]

##              NOG317682    NOG135470 NOG85325    NOG285859    NOG147792
## TARA_109_SRF         0 2.390962e-05        0 4.663604e-08 1.800215e-07
## TARA_149_MES         0 4.339824e-06        0 5.182915e-07 4.190123e-06
## TARA_110_MES         0 1.348252e-05        0 6.000043e-07 2.218342e-07
## TARA_102_MES         0 6.380711e-06        0 3.816016e-07 0.000000e+00
## TARA_142_SRF         0 9.484144e-06        0 6.437103e-08 1.132431e-06

Don’t hesitate to look the first rows of your data regularly using head(). It could be more convenient to display only the first 5 rows and columns when the data are big (tara_nog[c(1:5), c(1:5)]).

Practice:

Choose a dataset and load the data and the corresponding metadata into R.
How many different data type do you have?
How many samples do you have?
How many features do you have?
Is the data are in a good shape for the analysis?
Check the type of the metadata object.
Change it into data frame if it is necessary.
Check the column type of the metadata.
Change them if it is necessary.
How many type of variables (e.g. type of information) does the metadata contain?

These functions could help you: nrow(), ncol(), lapply(), dim(), t(), names(), type(), as.character() and unique().

4.3 Missing data

In the SNF paper, authors recommend to filter out samples with more than 20% of missing data in a certain data type. They also recommend to filter out features with more than 20% of missing data across samples. Then, they impute the remaining missing data using K nearest neighbors (KNN) imputation.

In this tutorial, we decide to remove samples with at least one missing data. To remove samples with missing data, we propose the following NARemoving() function. Input paramaters are:

data: the data type
margin: a vector giving the subscripts which the function will be applied over (e.g. 1 indicates rows and 2 indicates columns)
threshold: threshold above which samples/features are deleted

NARemoving <- function(data, margin, threshold){
    #' NA removing
    #'
    #' Calculate percentage of na
    #' Remove na from rows (margin = 1) or column (margin = 2)
    #' 
    #' @param data data.frame.
    #' @param margin int. 1 = row and 2 = column
    #' @param threshold int. Number of missing data accepted
    #'  
    #' @return Return data.frame with a specific number of na by row/column

  data_na <- apply(data, MARGIN = margin, FUN = function(v){sum(is.na(v)) / length(v) * 100})
  # print(table(data_na))
  toRemove <- split(names(data_na[data_na > threshold]), " ")[[1]]
  if(margin == 1){
    data_withoutNa <- data[!(row.names(data) %in% toRemove),]
    print(paste0("Remove ", as.character(length(toRemove)), " samples."))
  }
  if(margin == 2){
    data_withoutNa <- data[,!(colnames(data) %in% toRemove)]
    print(paste0("Remove ", as.character(length(toRemove)), " features"))
  }
  return(data_withoutNa)
}

For instance, the CLL drug data contain missing data (sample H024).

CLL_data_t$Drugs[c(1:5), c(1:5)]

##         D_001_1    D_001_2   D_001_3   D_001_4   D_001_5
## H045 0.02363938 0.04623274 0.3187471 0.8237027 0.8962777
## H109 0.07359900 0.10623002 0.2732891 0.7171379 0.8850003
## H024         NA         NA        NA        NA        NA
## H056 0.05813930 0.09022028 0.2322145 0.7225736 0.7957497
## H079 0.02042077 0.04750543 0.3638962 0.8073907 0.8794886

To remove samples with missing data in drug data, we use the following command line:

data = CLL_data_t$Drugs: remove the samples with missing data in the drug data
margin = 1: samples are in rows, so we want to apply the function on the rows
threshold = 0: we remove samples with at least one missing data

CLL_drug <- NARemoving(data = CLL_data_t$Drugs, margin = 1, threshold = 0)

## [1] "Remove 16 samples."

The H024 sample is not anymore in the CLL drug data.

CLL_drug[c(1:5), c(1:5)]

##         D_001_1    D_001_2   D_001_3   D_001_4   D_001_5
## H045 0.02363938 0.04623274 0.3187471 0.8237027 0.8962777
## H109 0.07359900 0.10623002 0.2732891 0.7171379 0.8850003
## H056 0.05813930 0.09022028 0.2322145 0.7225736 0.7957497
## H079 0.02042077 0.04750543 0.3638962 0.8073907 0.8794886
## H164 0.02962725 0.08054628 0.4725991 0.8179143 0.8927961

We repeat this step for each data type. Then, we filter out samples that are not present in all data type.

sampleNames <- Reduce(intersect, list(rownames(CLL_drug), rownames(CLL_mrna)))
CLL_drug <- CLL_drug[rownames(CLL_drug) %in% sampleNames,]
CLL_mrna <- CLL_mrna[rownames(CLL_mrna) %in% sampleNames,]

Practice:

Check if your samples are in rows and your features in columns.
Do you have missing data in your data?
Remove samples with at least one missing data.
Are you going to use all the data type available?

These functions could help you: is.na(), lapply(), table() and view().

4.4 Normalization and scaling

4.4.1 Normalization

The normalization should be adapted according the data type. For the data used here, we assumed that data have been already normalized, according their type. This step is really important, and should be correctly done before every type of data integration.

4.4.2 Scaling

Each feature (column) needs to have the mean equals to zero and the standard deviation equals to one. For that, the SNF package provides the function standardNormalization().

In the Figure 4.2, you can see the data distribution for the breast cancer miRNA data before (on the left) and after (on the right) scaling. After scaling, the data distribution should be normal.

hist(as.matrix(tcga_mirna), nclass = 100, main = "Breast cancer miRNA data", xlab = "values")
hist(as.matrix(tcga_mirna_scaled), nclass = 100, main = "Breast cancer miRNA scaled data", xlab = "values")

Figure 4.2: Breast cancer miRNA data distribution before (left) and after (right) scaling.

Practice:

Check if your samples are in rows and your features in columns.
Scale your data with standardNormalization().
Plot the data distribution before and after scaling for each data type.
What can you say about these distribution plots?

These functions could help you: hist(), as.matrix().

5 Similarity network

5.1 In summary

In this section, we create a sample network for each type of data, based on the similarity between samples. The main steps are (Figure 5.1):

Compute the distance between each pair of samples (distance matrix D).
Transform distances (from distance matrix D) into weights (into similarity matrix W) using distance with nearest neighbors.
Create the corresponding similarity network (G).

Similarity network creation overview. Data type 1 is in the first row, represented with the green matrices. Data type 2 is in the last row, represented with the purple matrices.

Figure 5.1: Similarity network creation overview. Data type 1 is in the first row, represented with the green matrices. Data type 2 is in the last row, represented with the purple matrices.

The similarity network is a sample network. It is defined as a network G with nodes (or vertices) called V and connections (or edges) called E. The connections between samples are weighted. Weights come from the similarity matrix.

We can define the similarity network G like this:

$G = (V, E, W)$

V: Nodes are samples
E: Edges are connections between samples
W: Edge weights are the similarity (or weight) between samples

5.2 Distance calculation

First, we calculate the distance between each pair of samples for each data type using the preprocessed data. The distance method used needs to be adapted to the feature type (e.g. continuous, discrete).

In the SNF paper (Wang et al.), authors suggest to use:

distance (e.g. euclidean) or correlation (e.g. pearson) for continuous features
chi-squared distance for discrete features
agreement based measure for binary features

In the SNF R package, the dist2() function performs a squared Euclidean distances between samples.

Practice:

Calculate the Euclidean distance between samples for each data type.
What are the dimensions of the created distance matrices?
Why is the diagonal close to zero?
What does a high distance between two samples mean?
What does a small distance mean?
Could you apply another distance calculation? Which ones?

These functions could help you: as.matrix(), dim(), nrow() and ncol().

5.3 Similarity calculation

The distance matrix D is then transformed into the similarity matrix W. Distances are converted to weights using the scaled exponential similarity kernel (µ) and average distances between samples and their nearest neighbors (ε):

\[ W = exp(-\frac{D^2}{µ\varepsilon})\] In other words, distances are converted to weights according the distance with the nearest neighbors of each sample pair.

The SNF R package proposes the affinityMatrix() to calculate this similarity matrix. In this case, affinity and similarity are equivalent. This function needs three parameters:

diff: distance matrix D
K: number of nearest neighbors (between 10 and 30)
sigma: hyperparameter or variance (between 0.3 and 0.8)

The K neighbors is used to set the similarities outside of the neighborhood to zero. The sigma parameter allows scaling exponential similarity kernel, which is used to calculated the similarity.

Then, you can visualized the similarity matrix using the pheatmap() function. This function creates an heatmap of samples. By default samples are clustered using hierarchical clustering. For a better visualization, we recommend to:

remove labels with show_rownames = FALSE and show_colnames = FALSE
give metadata to the parameter annotation
apply a log 10 transformation to the similarity matrix (log(x, 10))

Practice:

Calculate the similarity matrix using K = 20 and sigma = 0.5 for each data type.
Visualize the heatmap of the corresponding similarity matrices.
What does the red color mean? What does the blue color mean?
Are heatmaps similar between data types?
What can you say about these heatmaps?
How are samples relative to each other in the heatmap (e.g, close, group)?
Try with different parameters and visualize heatmaps. Can you explain what is happening?

6 Fusion

6.1 In summary

Previously, we created a similarity matrix W and its corresponding similarity network G for each data type (Figure 6.1).

In the previous step, we create a similarity matrix that contains weights for each data type. We also created the corresponding similarity network.

Figure 6.1: In the previous step, we create a similarity matrix that contains weights for each data type. We also created the corresponding similarity network.

Now, we integrate these similarity matrices (in the Figure 6.1 there are two data types). For that, we use an iterative fusion method. The number of iteration T, needs to be defined.

First, two matrices are created from the similarity matrix of each data type:

the P matrix: this matrix, also called status matrix, contains normalized weights (come from the similarity matrix W).
the S matrix: this matrix is the kernel matrix. It contains information about the nearest neighbors.

The P matrix carries the full information about the similarity of each samples to all others.

The S matrix carries the similarity to the K most similar samples for each sample (i.e. topology). The similarities between non-neighboring nodes are set to zero because authors assume that local similarities (high weights) are more reliable than the remote ones.

Then, the P matrix of each data type is iteratively updated with information from P matrices of the other data type, making them more similar at each step. In the Figure 6.2, you have an example with two data type. It’s a bit more complex with more data type (see the paper if you are interested in).

Example of the fusion method applied to two data types. Each data type is represented using a color: data type 1 in green and data type 2 in purple.

Figure 6.2: Example of the fusion method applied to two data types. Each data type is represented using a color: data type 1 in green and data type 2 in purple.

Finally after T iterations, the P matrices of each data type are merged together to create the final fused similarity matrix, and the corresponding fused similarity network.

Then, you can visualize the fused similarity network using Cytoscape.

6.2 Create the fused similarity matrix

First, we have to define the number of iteration called T. It should be between 10 and 20 (recommended by the authors).

Then, to perform the fusion, SNF R package proposes the SNF() function. You have to provide:

the list of similarity matrices of each data type
the number of nearest neighbors K (same as previously)
the number of iteration T

Practice:

Create the fused similarity matrix with 10 iterations (T = 10).
What are the dimensions of the fused similarity matrix?
What are values inside the fused similarity matrix?
How many values do you have in the fused similarity matrix?
How many values equal to zero do you have in the fused similarity matrix?
Visualize the corresponding heatmap.
What does the red color mean? What does the blue color mean?
Compare the fused similarity matrix heatmap and the data type heatmaps. What can you say?
How are samples relative to each other in the fused similarity matrix?
Try with another number of iteration. What’s happening?

These functions could help you: list(), length().

6.3 Visualize the fused similarity network

6.3.1 Create the fused similarity network

You can visualize the fused similarity network using Cytoscape.

First, you need to convert the fused similarity matrix into the corresponding fused similarity network. The igraph R package allows to create and manage networks.

You can create a the fused similarity network using the graph_from_adjacency_matrix() function. This function uses a similarity matrix to create the corresponding similarity network.

We don’t want duplicate information about connections between samples, neither connections between samples themselves (self loops). But we want to keep the connection weight values. You can use these parameters:

don’t take the diagonal: diag = FALSE
use only one part of the matrix: mode = "upper"
use the edge weights: weighted = TRUE

Then, you can save the fused similarity network into a edge file using write.table() function.

This is an example of the saving command line:

write.table(as_data_frame(W_net), "CLL_W_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

6.3.2 Visualize using Cytoscape

To create a network using Cytoscape, use the following steps:

6.3.2.1 Import files

Figure 6.3: Step 1 - Import files

Import Network from File: *_W_edgeList.txt and define:
- column 1 as Source node
- column 2 as Target node
- column 3 as Edge attributes
Import Table from File: *_metadata.txt (Import Data as Node Table Columns)

6.3.2.2 Network style

Figure 6.4: Step 2 - Change the network style

In the Style tab:

Fill Color
- select a column from the metadata. Here it’s growth_stage from the tomato dataset
- you can automatically fill the color by right click on Mapping Type tab
Shape = Ellipse
Lock node width and height

6.3.2.3 Network layout

Apply your favorite layout. For this step, we recommend to try at least yFiles Organic Layout.

6.3.3 Practice

Practice:

Create the fused similarity network with the fused similarity matrix.
Save the network into a file.
Visualize the fused similarity network in Cytoscape.
How many nodes do you have? How many edges do you have?
Why the edges number in the network is not the same as the number of values in the matrix?
Try different layouts.
What do you think about this network?

7 Threshold selection

7.1 In summary

In Cytoscape, you see that the fused similarity network is fully connected. Indeed, each sample is connected to all other samples. Remember, connections between samples are weighted: this weight represents the similarity between samples. A high weight value means a strong similarity between two samples whereas a low weight value means a weak similarity between two samples.

For a better visualization, we can select a threshold to decide which connections to keep, based on this weight value.

How to choose the good threshold? It’s an open question. There are some possibilities:

choose an arbitrary threshold (e.g. nearest neighbors)
choose a threshold based on basic metrics (e.g. median, quantiles)
choose a threshold using methods based on network topology
…

7.2 Select an arbitrary threshold

In this example, we select a threshold based on the weight distribution of the fused similarity network.

First, we extract edges from the network object. Indeed, the network object doesn’t contains duplicate edges, neither self loops unlike the corresponding fused similarity matrix W.

For instance, we extract weight values from the Tara Ocean fused similarity network. This network contains 9591 connections in total.

weights <- edge.attributes(tara_W_net)$weight

Then, we display the corresponding weight histogram in the Figure 7.1 (left). You can see a large number of small weight values. We can remove values between 0 and 0.01 where the majority of the small values seem to be. With the 0.01 threshold, 777 connections are kept.

In the Figure 7.1 (right), we log-transform weights. The weight distribution is binomial and we would like to cut the distribution into two parts. With the threshold 0.001 (log(0.001, 10) = -3), 6440 connections are kept.

## Raw weights
hist(weights, nclass = 100, main = "Fused similarity network weight distribution", xlab = "weights")
abline(v = 0.01, col = "red", lwd = 3)
## log10 weights
hist(log(weights, 10), nclass = 100, main = "Fused similarity network weight distribution", xlab = "log10(weights)")
abline(v = -3.1, col = "red", lwd = 3)

Weight distribution of the fused similarity network. On the left the weight distribution of the fused similarity network. On the right the log-transformed weight distribution of the fused similarity network.

Figure 7.1: Weight distribution of the fused similarity network. On the left the weight distribution of the fused similarity network. On the right the log-transformed weight distribution of the fused similarity network.

Advantages	Drawbacks
1. Easy to implement	1. Arbitrary
2. Fast	2. Doesn’t take account of the topology of the network
3. Visual

7.3 Select the threshold using quantiles

As example for this part, we select the median and the third quantile values as thresholds. We calculate the median and the third quantile of the weight values.

W_median <- median(x = weights)
W_q75 <- quantile(x = weights, 0.75)

The weight distribution and the log-transformed weight distribution are displayed in the Figure 7.2, respectively left and right.

## Raw weights
hist(weights, nclass = 100, main = "Fused similarity network weight distribution", xlab = "weights")
abline(v = W_median, col = "blue", lwd = 3)
text(W_median, 2000, pos = 4, "Median", col = "blue", cex = 1)
## log10 weights
hist(log(weights, 10), nclass = 100, main = "Fused similarity network weight distribution", xlab = "log10(weights)")
abline(v = log(W_median, 10), col = "blue", lwd = 3)
text(log(W_median, 10), 400, pos = 2, "Median", col = "blue", cex = 1)
abline(v = log(W_q75, 10), col = "purple", lwd = 3)
text(log(W_q75, 10), 350, pos = 4, "quantile 75%", col = "purple", cex = 1)

Figure 7.2: Weight distribution of the fused similarity network. On the left the weight distribution of the fused similarity network. On the right the log-transformed weight distribution of the fused similarity network.

The median is the value that splits data into two groups with the same number of data. With this value (0.002143), we selected 4795 connections.

The values above the third quantile value (0.0041194) are in the top of 25% highest weights. We selected 2398 connections.

Advantages	Drawbacks
1. Easy to implement	1. Arbitrary (but fitted to the data)
2. Fast	2. Doesn’t take account of the topology of the network
3. Fitted to the data

7.4 Threshold based on network topology

In the Zahoranszky-Kohalmi et al. paper, authors propose a method to determine the best threshold according to the topology of the network. This method calculates the Average Clustering Coefficient (ACC) for a whole network. The ACC represents a global parameter that characterizes the overall network topology.

As an Elbow approach, an ACC value is calculated for a range of thresholds. The obtained values are displayed and we choose the threshold that seems the best!

7.4.1 Functions

For help, we created two functions to calculate these ACC values and choose the best threshold based on the topology:

CCCalculation(): this function calculates the Clustering Coefficient (CC) for each node
ACCCalculation(); this function averages the CC for a network, in order to obtain the ACC

## CC calculation function
CCCalculation <- function(node, graph){
    #' Clustering Coefficient (CC) calculation
    #'
    #' Calculate the Clustering Coefficient (CC) for each node in a network
    #' 
    #' @param node str.
    #' @param graph igraph. Network object (e.g. the fused network object)
    #'  
    #' @return Return the corresponding CC value
    
  degNode <- degree(graph = graph, v = node, loops = FALSE)
  if(degNode > 1){
    neighborNames <- neighbors(graph = graph, v = node)
    graph_s <- subgraph(graph = graph, vids = neighborNames)
    neighborNb <- sum(degree(graph_s, loops = FALSE))
    CC <- neighborNb / (degNode * (degNode-1))
  }else{CC <- 0}
  return(CC)
}

## ACC calculation function
ACCCalculation <- function(graph){
    #' Average Clustering Coefficient (ACC) calculation
    #'
    #' It average the Clustering Coefficient (CC) of a network
    #' 
    #' @param graph igraph. Network object (e.g. the fused network object)
    #'  
    #' @return Return the corresponding ACC value
    
  nodes <- V(graph)
  ACC <- do.call(sum, lapply(nodes, CCCalculation, graph)) / length(nodes)
  return(ACC)
}

7.4.2 Range of thresholds

First, we determine a range of thresholds. The ACC precision increases with the number of chosen thresholds (choose at least 100).

summary(weights)

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 5.696e-05 4.389e-04 2.143e-03 3.623e-03 4.119e-03 1.414e-01

thresholds <- seq(0, 0.1, 0.0005)

7.4.3 Calculate the ACC values

Calculate the ACC values for each selected threshold.

ACC_W <- do.call(rbind, lapply(thresholds, function(t, net){
  net_sub <- subgraph.edges(net, E(net)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(net_sub), "thresholds" = t, "EN" = length(E(net_sub)))
  return(df)
}, tara_W_net))

The ACC values are displayed in the Figure 7.3 (left). The number of network edges for each threshold is displayed in the Figure 7.3 (right).

plot(x = ACC_W$thresholds, y = ACC_W$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC calculation of the Fused network W", type = "o")
points(x = ACC_W$thresholds[1], y = ACC_W$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = ACC_W$thresholds[21], y = ACC_W$ACC[21], col = "pink", pch = 16, cex = 1.2)
points(x = ACC_W$thresholds[29], y = ACC_W$ACC[29], col = "purple", pch = 16, cex = 1.2)
abline(v = ACC_W$thresholds[29], col = "purple")
text(ACC_W$thresholds[29], 0.7, pos = 4, paste0("Threshold = ",  ACC_W$thresholds[29]), col = "purple")
text(ACC_W$thresholds[29], 0.6, pos = 4, paste0("ACCmax = ",  round(ACC_W$ACC[29], 2)), col = "purple")
plot(x = ACC_W$thresholds, y = ACC_W$EN, xlab = "thresholds", ylab = "number of edges", main = "EN of the Fused network W", type = "o")
abline(v = ACC_W$thresholds[29], col = "purple")
text(ACC_W$thresholds[29], 2800, pos = 4, paste0("Threshold = ",  ACC_W$thresholds[29]), col = "purple")
text(ACC_W$thresholds[29], 2000, pos = 4, paste0("ACCmax = ",  round(ACC_W$ACC[29], 2)), col = "purple")
text(ACC_W$thresholds[29], 1200, pos = 4, paste0("EN = ",  ACC_W[29, "EN"]), col = "purple")

Figure 7.3: Left: Average Clustering Coeeficient (ACC) values for each threshold - Right: Number of edges on network for each threshold

On the Figure 7.3 you can see a peak that stands out in comparison with the others.

Red dot: no filter, every sample is connected to each other
Pink dot: the smallest value before the peak
Purple dot: the local maxima of ACC (we want this one)

Advantages	Drawbacks
1. Take account of the topology of the network	1. Might be time consuming
	2. Interpretation (need to be used to this method)
	3. Sometimes, no peak …

7.5 Visualize this threshold in Cytoscape

The fused similarity network is already imported in Cytoscape. You already changed the network style and the network layout. If it is not, you can go to the previous Cytoscape section.

We have a fully connected network. To select edges based on their weights, use the following steps:

7.5.1 Filter column

Figure 7.4: Filter edge weight using the determined threshold.

In the Filter tab:

Add a new condition
Select Column Filter
Choose the Edge: weight column name
Put your threshold in the range

7.5.2 Create a new network visualization

Figure 7.5: Create new network visualization according the determined threshold

Select All Nodes
Create a New Network from Selected Nodes, Selected Edges
Apply your favorite layer on this new network

7.6 Practice

Now, it’s your turn to find the best threshold!!

Practice:

Determine the threshold:

1.1. Select a threshold based on the weight distribution.

1.2. Use the median as threshold.

1.3. Determine the threshold based on the topology of the network.
Visualize these thresholds in Cytoscape.
Determine for each network that are in Cytoscape:

3.1. Number of edges and nodes.

3.2. Number of isolated samples.
For you, which threshold is better?
For each data type:

5.1. Create a network from the similarity matrix

5.2. Do the 1-4 steps

Keep in mind that you can change this threshold in Cytoscape anytime.

8 Downstream analysis

You integrated your data and created the corresponding fused similarity network: CONGRATULATIONS!!

This fused similarity network can be use for downstream analysis based on network’s algorithms such as clustering, retrieval or classification. You can also visualize the network and add some external data.

In this hands-on, we give you two examples to illustrate what you can do with the fused similarity network: clustering and visualization using Cytoscape.

8.1 Clustering

In the SNF paper, authors propose a clustering method called spectralClustering(). This function takes in input three parameters:

affinity: fused similarity matrix W
K: number of clusters we want
type: the variants of spectral clustering to use (default 3)

Samples are clustered together according to their similarity. The number of clusters needs to be defined. Obviously, this cluster number can be chosen using an Elbow approach, which would be less arbitrary.

First, define the number of clusters. Here, we define four clusters.

C <- 4

Then, perform the clustering. Results are stored into the group variable.

group <- data.frame(Groups = spectralClustering(tara_W, C)) 
row.names(group) <- colnames(tara_W)

Next, merge the clustering results with the metadata.

dataGroups <- merge(metadata, group, by = 0) 
head(dataGroups)

##      Row.names ocean depth Groups
## 1 TARA_004_DCM   NAO   DCM      1
## 2 TARA_004_SRF   NAO   SRF      1
## 3 TARA_007_DCM    MS   DCM      1
## 4 TARA_007_SRF    MS   SRF      1
## 5 TARA_009_DCM    MS   DCM      1
## 6 TARA_009_SRF    MS   SRF      1

Finally, save the merged data into a file.

write.table(dataGroups, "TARAOcean_4clusters.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

Practice:

Choose three cluster numbers (e.g. two, three and nine)
Perform the clustering.
Save results into file.

You might merge clustering results to create on result file using merge() function.

8.2 Visualization on Cytoscape

As a reminder:

How to import files and change network visualization style, see section 6.3.2
How to apply a threshold on edges, see section 7.5.2

You already loaded the fused similarity network in Cytoscape. And you also removed the weak edges. To visualize the clustering results follow the steps:

Figure 8.1: Import table to the Network Collection.

Import Table from File: *_clusters.txt to the Network Collection (information will be add to every network in the collection)
Color the nodes according to their cluster
Create the corresponding legend using the Legend Panel tab
- First, Scan Network
- Then, modify the title and the subtitle
- Finally, Refresh Legend
To move legend elements, click on the T icon

Figure 8.2: Create a legend using the Legend Panel.

This is a network visualization example for each dataset:

$Example of fused similarity network visualization using Cytoscape for each dataset.$

Figure 8.3: Example of fused similarity network visualization using Cytoscape for each dataset.

Practice:

Import the clustering results file.
Change the style of the network.
Try to have a similar network as you can see in the figure.
How can you interpret the results?
Play with the colors, change the metadata used and do what you want with Cytoscape!

9 Go further

Now, you know how to integrate different type of data using the Similarity Network Fusion (SNF) method.

You’re aware of the important steps such as the preprocessing of the data (e.g. normalization and scaling, data shape), the similarity network creation and the fusion.

You also see how to perform clustering on the fused similarity network and how to visualize results.

This part is to go further in your analysis and your interpretation of the results.

9.1 Normalization and distance calculation

At the beginning of the hands-on, we assumed that data have been already well normalized. We also used only the euclidean distance, whatever the types of data.

It could be interesting to try different normalization and distance calculation to choose the most-adapted preprocessing of the data.

For instance, the two main adapted normalization methods for the Tara Ocean data are:

Centered log ratio method (CLR)
Total sum-scaling (TSS)

Moreover, other distance distances exist and are more appropriate for ecological data than Euclidean distance:

Bray-Curtis distance
Hellinger distance

These functions could help you: distance(), tss() or clr(). They are available in the ecodist, hilldiv and compositions packages.

Practice:

Try another normalization method.
Try another distance calculation or correlation.
Run a new SNF analysis.
Interpret the results.

9.2 Data type contribution

Did you notice the edge colors in the Figure 8.3? Colors represent the contribution of each data type in each sample association.

In the SNF paper, authors did like this:

The edge is considered supported by a single data if its weight in that data type’s network is more than 10% higher than the weight of the same edge in the other data type’s networks.
If the difference between two highest edge weights from the corresponding data types is less than 10%, the edge is considered supported by those 2 data types.
If the difference is less than 10% between all data type, the edge is supported by all data types.

Depending of the number of the integrated data type, we suggest two workflows:

9.2.1 Data type contribution: with 2 data types

Transform similarity network into data frames (to transform similarity matrix into similarity network see section 6.3.1).

tara_W_df <- as_data_frame(tara_W_net)
tara_nog_df <- as_data_frame(tara_nog_net)
tara_phy_df <- as_data_frame(tara_phy_net)

Merge data frames into one data frame:

tmp <- merge(tara_nog_df, tara_phy_df, by = c(1,2), suffixes = c("_tara_nog", "_tara_phy"))
weights_df <- merge(tmp, tara_W_df, by = c(1,2))
head(weights_df)

##           from           to weight_tara_nog weight_tara_phy       weight
## 1 TARA_004_DCM TARA_138_DCM    5.941851e-04    2.261158e-04 0.0093268270
## 2 TARA_004_SRF TARA_004_DCM    1.514222e-03    2.973412e-04 0.0151933836
## 3 TARA_004_SRF TARA_133_MES    7.162040e-06    1.011284e-05 0.0001358067
## 4 TARA_004_SRF TARA_138_DCM    8.719842e-05    1.805816e-04 0.0028421617
## 5 TARA_007_DCM TARA_004_DCM    1.163166e-05    3.450973e-05 0.0030039624
## 6 TARA_007_DCM TARA_004_SRF    1.160152e-05    3.159833e-05 0.0025836564

Determine data type contribution on each edge:

sources_df <- as.data.frame(do.call(rbind, apply(weights_df, 1, function(s){
  sources <- as.numeric(s[c(3,4)])
  ## Initiate keep vector with FALSE value
  keep <- c(rep(FALSE, length(sources)))
  ## Where is the max value ?
  max_ind <- which.max(sources)
  ## Keep it 
  max <- max(sources)
  keep[max_ind] <- TRUE
  
  ## Retrieved indices of all non max value
  tmp <- seq(1, length(sources))
  tmp <- tmp[!tmp %in% max_ind]
  ## If max weight < 10% of other weight, put it to TRUE
  for(i in tmp){
    w <- as.numeric(sources[i])
    if(max > w+(w*10/100)){keep[i] <- FALSE}
    else{keep[i] <- TRUE}
  }
  ## Select all the TRUE weights
  ## Compare them by pairs
  ## If w1 - w2 is > 10%, they are not selected anymore
  ## Except if one of them is the max value
  keep_df <- as.data.frame(table(keep))
  if(keep_df[keep_df$keep == TRUE, "Freq"] > 1){
    ind <- which(keep == TRUE)
    ind_comb <- combn(x = ind, m = 2)
    for(i in seq(1:ncol(ind_comb))){
      vector <- ind_comb[,i]
      v1 <- as.numeric(sources[vector[1]])
      v2 <- as.numeric(sources[vector[2]])
      if(abs(v1 - v2) < (v1*10/100) && abs(v1 - v2) < (v2*10/100)){
        keep[vector[1]] <- TRUE
        keep[vector[2]] <- TRUE
      }else{
        keep[vector[1]] <- FALSE
        keep[vector[2]] <- FALSE
        keep[max_ind] <- TRUE
      }
    }
  }
  names(keep) <- names(s[c(3,4)])
  final <- c(s[1], s[2], keep)
  return(final)
}, simplify = FALSE)))

Encode data type contribution to help the visualization:

data1 (tara_nog) = 1
data2 (tara_phy) = 2
data1 + data2 = 3

sources_df$tara_nog <- ifelse(sources_df$tara_nog == TRUE, 1, 0)
sources_df$tara_phy <- ifelse(sources_df$tara_phy == TRUE, 2, 0)
sources_df$sum <- apply(sources_df[,c(3,4)], 1, sum)

Write results into a file:

tara_W_df_withSource <- (merge(W_df, sources_df, by = c(1,2)))
write.table(tara_W_df_withSource, "TaraOcean_W_edgeList_withSource.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

Counts number of each combination of data:

as.data.frame(table(sources_df$sum))

Finally, load this file into Cytoscape and change the edge colors (see section 6.3.2 to import files and change node color).

9.2.2 Data type contribution: with 3 data types

Transform similarity network into data frames (to transform similarity matrix into similarity network see section 6.3.1).

tcga_W_df <- as_data_frame(tcga_W_net)
tcga_mirna_df <- as_data_frame(tcga_mirna_net)
tcga_mrna_df <- as_data_frame(tcga_mrna_net)
tcga_prot_df <- as_data_frame(tcga_prot_net)

Merge data frames into one data frame:

tmp1 <- merge(tcga_mirna_df, tcga_mrna_df, by = c(1,2), suffixes = c("_tcga_miRNA", "_tcga_mRNA"))
tmp2 <- merge(tcga_prot_df, tcga_W_df, by = c(1, 2), suffixes = c("_tcga_prot", "")) 
weights_df <- merge(tmp1, tmp2, by = c(1,2))
head(weights_df)

##   from   to weight_tcga_miRNA weight_tcga_mRNA weight_tcga_prot      weight
## 1 A03L A08O      3.664207e-05     4.127975e-04     1.376957e-03 0.007833912
## 2 A03L A0BS      9.365460e-05     8.151721e-05     3.543471e-04 0.003016587
## 3 A03L A0E1      7.392441e-05     3.924428e-04     4.981989e-04 0.004073567
## 4 A03L A0FS      1.971148e-05     1.154763e-04     5.335656e-04 0.001935859
## 5 A03L A0H7      2.216591e-04     3.549924e-04     4.890533e-05 0.005945000
## 6 A03L A0W4      6.924449e-05     2.586121e-04     8.181917e-04 0.002147077

Determine data type contribution for each edge:

sources_df <- as.data.frame(do.call(rbind, apply(weights_df, 1, function(s){
  sources <- as.numeric(s[c(3,4,5)])
  ## Initiate keep vector with FALSE value
  keep <- c(rep(FALSE, length(sources)))
  ## Where is the max value ?
  max_ind <- which.max(sources)
  ## Keep it 
  max <- max(sources)
  keep[max_ind] <- TRUE
  
  ## Retrieved indices of all non max value
  tmp <- seq(1, length(sources))
  tmp <- tmp[!tmp %in% max_ind]
  ## If max weight < 10% of other weight, put it to TRUE
  for(i in tmp){
    w <- as.numeric(sources[i])
    if(max > w+(w*10/100)){keep[i] <- FALSE}
    else{keep[i] <- TRUE}
  }
  ## Select all the TRUE weights
  ## Compare them by pairs
  ## If w1 - w2 is > 10%, they are not selected anymore
  ## Except if one of them is the max valye
  keep_df <- as.data.frame(table(keep))
  if(keep_df[keep_df$keep == TRUE, "Freq"] > 1){
    ind <- which(keep == TRUE)
    ind_comb <- combn(x = ind, m = 2)
    for(i in seq(1:ncol(ind_comb))){
      vector <- ind_comb[,i]
      v1 <- as.numeric(sources[vector[1]])
      v2 <- as.numeric(sources[vector[2]])
      if(abs(v1 - v2) < (v1*10/100) && abs(v1 - v2) < (v2*10/100)){
        keep[vector[1]] <- TRUE
        keep[vector[2]] <- TRUE
      }else{
        keep[vector[1]] <- FALSE
        keep[vector[2]] <- FALSE
        keep[max_ind] <- TRUE
      }
    }
  }
  names(keep) <- names(s[c(3,4,5)])
  final <- c(s[1], s[2], keep)
  return(final)
}, simplify = FALSE)))

Encode data type contribution to help the visualization:

data1 (tcga_mirna) = 1
data2 (tcga_mrna) = 2
data3 (tcga_prot) = 4
data1 + data2 = 3
data1 + data3 = 5
data2 + data3 = 6
data1 + data2 + data3 = 7

sources_df$weight_tcga_miRNA <- ifelse(sources_df$weight_tcga_miRNA == TRUE, 1, 0)
sources_df$weight_tcga_mRNA <- ifelse(sources_df$weight_tcga_mRNA == TRUE, 2, 0)
sources_df$weight_tcga_prot <- ifelse(sources_df$weight_tcga_prot == TRUE, 4, 0)
sources_df$sum <- apply(sources_df[,c(3,4,5)], 1, sum)

Write results into a file:

W_df_withSource <- (merge(tcga_W_df, sources_df, by = c(1,2)))
write.table(W_df_withSource, "TCGA_W_edgeList_withSource.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

Counts number of each combination of data:

as.data.frame(table(sources_df$sum))

##   Var1 Freq
## 1    1 3051
## 2    2 2115
## 3    3  280
## 4    4 5090
## 5    5  317
## 6    6  289
## 7    7   33

Finally, load this file into Cytoscape and change the edge colors (see section 6.3.2 to import files and change node color).

Practice:

Determine the data type contribution for each edge.
How many each data type support edges?
Save the results into a file.
Load this file into Cytoscape.
Change the edge style.
What do you notice in the network?

10 Tara Ocean dataset

10.1 Input data

The metagenomic dataset from the Tara Ocean project contains 2 data types:

orthologous genes: the relative abundance of groups of orthologous genes (OGs)
phylogenetic profil: the counts of S16 rRNA

Data files are available in the summer school’s GitHub repository:

10.1.1 Load dataset

First, we load the data type from TARAoceans_proNOGS.cvs and TARAoceans_proPhylo.csv files.

10.1.1.1 Orthologous genes (nog) data

The data file contains header (head = TRUE) and the first column contains row names (row.names = 1). Below, the first rows and columns are displayed.

tara_nog <- read.table(file = "../00_Data/TaraOcean_mibiomics/TARAoceans_proNOGS.csv", sep = ",", head = TRUE, row.names = 1)
tara_nog[c(1:5), c(1:5)]

##              NOG317682    NOG135470 NOG85325    NOG285859    NOG147792
## TARA_109_SRF         0 2.390962e-05        0 4.663604e-08 1.800215e-07
## TARA_149_MES         0 4.339824e-06        0 5.182915e-07 4.190123e-06
## TARA_110_MES         0 1.348252e-05        0 6.000043e-07 2.218342e-07
## TARA_102_MES         0 6.380711e-06        0 3.816016e-07 0.000000e+00
## TARA_142_SRF         0 9.484144e-06        0 6.437103e-08 1.132431e-06

Dimensions of the data.

dim(tara_nog)

## [1] 139 638

The nog data contain 139 samples (rows) and 638 features (columns). Data are in the right shape: samples in rows and features in columns.

10.1.1.2 Phylogentic profil (phy) data

The data file contains header (head = TRUE) and the first column contains row names (row.names = 1). Below, the first rows and columns are displayed.

tara_phy <- read.table(file = "../00_Data/TaraOcean_mibiomics/TARAoceans_proPhylo.csv", sep = ",", head = TRUE, row.names = 1)
tara_phy[c(1:5), c(1:3)]

##              EU638706.1.1353 JN537192.1.1500 CAFJ01000195.23.1515
## TARA_109_SRF               0               0                    0
## TARA_149_MES               0               0                    0
## TARA_110_MES               0               4                    0
## TARA_102_MES               0               1                    0
## TARA_142_SRF               0               3                    0

Number of rows in the data.

nrow(tara_phy)

## [1] 139

Number of columns in the data.

ncol(tara_phy)

## [1] 356

The phy data contain 139 samples (rows) and 356 features (columns). Data are in the right shape: samples in rows and features in columns.

10.1.2 Load metadata

The TARAoceans_metadata.csv file contains metadata. It contains header (head = TRUE) and row names (row.names = 1).

tara_metadata <- read.table(file = "../00_Data/TaraOcean_mibiomics/TARAoceans_metadata.csv", sep = ",", head = TRUE, row.names = 1)
head(tara_metadata)

##              ocean depth
## TARA_109_SRF   SPO   SRF
## TARA_149_MES   NAO   MES
## TARA_110_MES   SPO   MES
## TARA_102_MES   SPO   MES
## TARA_142_SRF   NAO   SRF
## TARA_109_DCM   SPO   DCM

The metadata contains two types of information: ocean and depth.

names(tara_metadata)

## [1] "ocean" "depth"

Samples come from 8 oceans.

unique(tara_metadata$ocean)

## [1] "SPO" "NAO" "IO"  "SO"  "SAO" "NPO" "RS"  "MS"

Samples were collected in 4 different depths.

unique(tara_metadata$depth)

## [1] "SRF" "MES" "DCM" "MIX"

10.1.3 Missing data

The data don’t contain missing values. We can go to the following steps.

table(is.na(tara_nog))

## 
## FALSE 
## 88682

table(is.na(tara_phy))

## 
## FALSE 
## 49484

10.1.4 Scaling

We assume that data have been already prepared and normalized.

10.1.4.1 nog data

Nog data are scaled: each column will scaled to have the mean equals to zero and the standard deviation equals to one.

tara_nog_scaled <- standardNormalization(tara_nog)

The following figures are the distribution of the data, before (left) and after (right) scaling. We expected a normal distribution of the data after scaling.

hist(as.matrix(tara_nog), nclass = 100, main = "Orthologous genes - Prepared data", xlab = "values")
hist(tara_nog_scaled, nclass = 100, main = "Orthologous genes - Scaled data", xlab = "values")

Before scaling, data values are almost all closed to zero. After scaling, data values seem to follow something close to a normal distribution. Data values are centered to zero.

10.1.4.2 phy data

Phy data are scaled: each column will scaled to have the mean equals to zero and the standard deviation equals to one.

tara_phy_scaled <- standardNormalization(tara_phy)

The following figures are the distribution of the data, before (left) and after (right) scaling. We expected a normal distribution of the data after scaling.

hist(as.matrix(tara_phy), nclass = 100, main = "Phylogenetic profile - Prepared data", xlab = "values")
hist(tara_phy_scaled, nclass = 100, main = "Phylogenetic profile - Scaled data", xlab = "values")

This kind of data are sparse: there are a lot of zero values. Majority of the values are in the first range on the histogram. After scaling, data values seem to follow something close to a normal distribution. Data values are centered to zero.

10.2 Similarity network

In this part, we create the similarity network for each data type.

10.2.1 Distance calculation

We calculate the Euclidean distance between each pair of samples for each type of data.

tara_nog_dist <- dist2(tara_nog_scaled, tara_nog_scaled)
tara_phy_dist <- dist2(tara_phy_scaled, tara_phy_scaled)

The distance matrix dimensions are 139 rows and 139 columns. Indeed, we calculated pairwise distance, so the matrix contains samples in rows and in columns.

dim(tara_nog_dist)

## [1] 139 139

The diagonal is composed of zero values (or values very closed). Indeed, there is no distance between the same sample.

tara_nog_dist[c(1:5), c(1:5)]

##              TARA_109_SRF TARA_149_MES TARA_110_MES TARA_102_MES TARA_142_SRF
## TARA_109_SRF 1.705303e-13     536.8095     329.8441 5.991295e+02     646.4804
## TARA_149_MES 5.368095e+02       0.0000     261.2395 3.325547e+02     738.4168
## TARA_110_MES 3.298441e+02     261.2395       0.0000 2.078134e+02     638.5818
## TARA_102_MES 5.991295e+02     332.5547     207.8134 2.273737e-13     877.6855
## TARA_142_SRF 6.464804e+02     738.4168     638.5818 8.776855e+02       0.0000

High distance values mean that samples are not similar. And small distance values mean that samples are similar.

10.2.2 Similarity calculation

The distance matrix is then transformed into similarity matrix for each data type. We set two parameters:

K = 20: number of nearest neighbors
sigma = 0.5: hyperparameter

K <- 20
signma <- 0.5

The affinityMatrix() function transforms the distance into similarity according the distance with the nearest neighbors.

tara_nog_W <- affinityMatrix(tara_nog_dist, K, signma)
tara_phy_W <- affinityMatrix(tara_phy_dist, K, signma)

The following figures are the heatmap of the similarity matrix (W) of each data type. The left heatmap are the nog data and the right heatmap are the phy data. Samples are clustered using hierarchical clustering. For a better visualization, we log-transform similarities.

pheatmap(log(tara_nog_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = tara_metadata, main = "Orthologous genes - log10-transformed similarity values")
pheatmap(log(tara_phy_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = tara_metadata, main = "Phylogenetic profil - log10-transformed similarity values")

Red color means a high similarity value between two samples whereas blue color means a small similarity value between two samples.

The two heatmaps are different. Data seem to probably carry different type of information about the samples:

for the nog data (left), we can see two main groups of samples. Each group seems to be composed of the same ocean depth. It’s not clear for the ocean regions.
for the phy data (right), we do not identify clear groups. But, samples seem to be group by the depth too.

10.3 Fusion

We created a similarity matrix for each data type. We saw that each network carries common information and its own information. Now, we will integrate all this information into only one fused similarity matrix.

10.3.1 Create the fused similarity matrix

We create the fused similarity matrix using these three parameters:

list that contains the nog and phy similarity matrices
K = 20: number of the nearest neighbors
T = 10: number of iterations

K <- 20
T <- 10

tara_W <- SNF(list(tara_nog_W, tara_phy_W), K, T)
tara_W[c(1:5), c(1:5)]

##              TARA_109_SRF TARA_149_MES TARA_110_MES TARA_102_MES TARA_142_SRF
## TARA_109_SRF 5.000000e-01 9.778842e-05 0.0002535809 0.0004410605 0.0019200171
## TARA_149_MES 9.778842e-05 5.000000e-01 0.0129882372 0.0126485110 0.0004977051
## TARA_110_MES 2.535809e-04 1.298824e-02 0.5000000000 0.0266185584 0.0029147232
## TARA_102_MES 4.410605e-04 1.264851e-02 0.0266185584 0.5000000000 0.0009179439
## TARA_142_SRF 1.920017e-03 4.977051e-04 0.0029147232 0.0009179439 0.5000000000

The dimension of the fused similarity matrix are 139 rows and 139 columns, such as the previous similarity matrices. The fused similarity matrix contains similarities between samples, we can also called them weights.

The fused similarity matrix contains 19321 weights.

length(tara_W)

## [1] 19321

The fused similarity matrix doesn’t contain zero:

length(tara_W[length(tara_W) == 0])

## [1] 0

The following figure is the heatmap of the fused similarity matrix. Samples are automatically clustered with a hierarchical clustering. Weights are log-transformed for a better visualization.

pheatmap(log(tara_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = tara_metadata, main = "Tara Ocean data - log10-transformed fused similarity matrix")

Groups are more clearly defined in this fused similarity matrix. One corresponds to the depth MESO and other to a mix of DCM and SRF. The fused similarity matrix seems to be a mix of each similarity data type matrix.

10.3.2 Visualized the fused similarity network

Now, we create a fused similarity network from the fused similarity matrix. Self loops are remove (diag = FALSE) and only the upper values of the matrix are taken (mode = "upper", avoid duplicate information).

tara_W_net <- graph_from_adjacency_matrix(tara_W, weighted = TRUE, mode = "upper", diag = FALSE)

Then, the fused similarity network is saved into a the TaraOcean_W_edgeList.txt file:

write.table(as_data_frame(tara_W_net), "../02_Results/01_TaraOcean/TaraOcean_W_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

This files is loaded into Cytoscape. The Figure 10.1 shows the fused similarity network of the Tara Ocean dataset.

Figure 10.1: The fused similarity network of the Tara Ocean dataset.

According to Cytoscape, the network contains 139 samples (nodes) and 9591 connections (edges). The number of edges is smaller than in the similarity matrix because in the similarity matrix the weights are duplicates. The similarity matrix contains also the weights for each sample compare to itself (self loops).

For now, the network is fully connected: each sample is connected to every sample. Connections between samples are weights: some connections are strong (samples are similar) some other are weak (samples are not similar).

10.4 Threshold selection

So in this section, we will choose a threshold to keep the strongest connections.

10.4.1 Fused similarity network

10.4.1.1 Arbitrary threshold

We extract the weight to display the corresponding distribution to try to find a threshold. The distribution in the left is created using the raw weights. The distribution in the right is created using the log-transformed weights.

tara_weights <- edge.attributes(tara_W_net)$weight
hist(tara_weights, nclass = 100, main = "Fused similarity network weight distribution", xlab = "weights")
hist(log(tara_weights, 10), nclass = 100, main = "Fused similarity network weight distribution", xlab = "weights")
abline(v = log(0.001, 10), col = "cyan", lwd = 3)

The log-transformed weight distribution shows a binomial distribution. We would probably like to cut between the two peaks and choose the corresponding weight: 0.001 With this threshold, we select 6440 connections.

10.4.1.2 Mean and third quantile

We calculate the median of the weights.

tara_W_median <- median(x = tara_weights)
tara_W_median

## [1] 0.002143026

With the mean (0.002143) as threshold, we select 4796 connections.

length(tara_weights[tara_weights>=tara_W_median])

## [1] 4796

Calculate the third quantile of the weights:

tara_W_q75 <- quantile(x = tara_weights, 0.75)
tara_W_q75

##         75% 
## 0.004119391

With the third quantile (0.0041194) as threshold, we select 2398 connections.

length(tara_weights[tara_weights>=tara_W_q75])

## [1] 2398

In the following figure, we display the log-transformed weight distribution with the two previous calculated metrics as markers.

hist(log(tara_weights, 10), nclass = 100, main = "Fused similarity network weight distribution", xlab = "log10(weights)")
abline(v = log(tara_W_median, 10), col = "blue", lwd = 3)
text(log(tara_W_median, 10), 400, pos = 2, "Median", col = "blue", cex = 1)
abline(v = log(tara_W_q75, 10), col = "purple", lwd = 3)
text(log(tara_W_q75, 10), 400, pos = 4, "quantile 75%", col = "purple", cex = 1)

10.4.1.3 Topology network

To determine a range of thresholds to try, we check the weights.

summary(c(tara_W))

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 0.0000570 0.0004456 0.0021613 0.0071942 0.0042193 0.5000000

We set the range between 0 and 0.0005.

thresholds <- seq(0, 0.1, 0.0005)
length(thresholds)

## [1] 201

Then, we calculate the Average Clustering Coefficient for each threshold.

tara_W_ACC <- do.call(rbind, lapply(thresholds, function(t, net){
  net_sub <- subgraph.edges(net, E(net)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(net_sub), "thresholds" = t, "EN" = length(E(net_sub)))
  return(df)
}, tara_W_net))

Calculated values are displayed in the following figures.

plot(x = tara_W_ACC$thresholds, y = tara_W_ACC$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC calculation of the Fused network W", type = "o")
points(x = tara_W_ACC$thresholds[1], y = tara_W_ACC$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = tara_W_ACC$thresholds[21], y = tara_W_ACC$ACC[21], col = "pink", pch = 16, cex = 1.2)
points(x = tara_W_ACC$thresholds[29], y = tara_W_ACC$ACC[29], col = "purple", pch = 16, cex = 1.2)
abline(v = tara_W_ACC$thresholds[29], col = "purple")
text(tara_W_ACC$thresholds[29], 0.7, pos = 4, paste0("Threshold = ",  tara_W_ACC$thresholds[29]), col = "purple")
text(tara_W_ACC$thresholds[29], 0.6, pos = 4, paste0("ACCmax = ",  round(tara_W_ACC$ACC[29], 2)), col = "purple")
plot(x = tara_W_ACC$thresholds, y = tara_W_ACC$EN, xlab = "thresholds", ylab = "number of edges", main = "EN of the Fused network W", type = "o")
abline(v = tara_W_ACC$thresholds[29], col = "purple")
text(tara_W_ACC$thresholds[29], 2800, pos = 4, paste0("Threshold = ",  tara_W_ACC$thresholds[29]), col = "purple")
text(tara_W_ACC$thresholds[29], 2000, pos = 4, paste0("ACCmax = ",  round(tara_W_ACC$ACC[29], 2)), col = "purple")
text(tara_W_ACC$thresholds[29], 1200, pos = 4, paste0("EN = ",  tara_W_ACC[29, "EN"]), col = "purple")

Figure 10.2: Left: Average Clustering Coeeficient (ACC) values for each threshold - Right: Number of edges on network for each threshold

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.

If we selected the purple local maxima, we will have 404 edges. It could be not enough edges. Let’s see during the visualization.

tara_W_ACC$thresholds[29]

## [1] 0.014

10.4.1.4 Visualization using Cytoscape

The network visualization on the left was created with the third quantile (0.0041194). The network visualization on the right was created with the ACC method (0.014).

Figure 10.3: The fused network of the Tara Ocean dataset.

The left network contains lot of edges and it’s difficult to see clear connection between samples. Nevertheless, we can see two groups of samples. It could be interesting to color the nodes with the depth information.

This trend is also shows in the right network. Moreover, ocean samples seem to be groups together.

Network visualizations are available in the TARAocean_cytoscape.cys file.

10.4.2 Orthologous gene data

Similarity matrix is transformed to a similarity network and saved into a file. We still keep only half (mode = "upper") of the similarity matrix (avoid redundancies) and remove the self loop (diag = FALSE).

tara_nog_net <- graph_from_adjacency_matrix(tara_nog_W, weighted = TRUE, mode = "upper", diag = FALSE)
write.table(as_data_frame(tara_nog_net), "../02_Results/01_TaraOcean/TaraOcean_nog_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

10.4.2.1 Arbitrary threshold

tara_weights <- edge.attributes(tara_nog_net)$weight
hist(tara_weights, nclass = 100, main = "nog similarity network weight distribution", xlab = "weights")
hist(log(tara_weights, 10), nclass = 100, main = "nog similarity network weight distribution", xlab = "weights")
abline(v = log(1.584893e-05, 10), col = "cyan", lwd = 3)

The log-transformed weight distribution shows a kind of normal distribution. We would probably like to cut in the middle of the peak, or just before or after. If we cut in the middle, the corresponding weight is: 1.584893e-05. With this threshold, we select 6666 connections.

10.4.2.2 Mean and third quantile

Calculate the median.

tara_nog_median <- median(x = tara_weights)
tara_nog_median

## [1] 2.876783e-05

Number of selected edges with the median as threshold.

length(tara_weights[tara_weights >= tara_nog_median])

## [1] 4796

Calculate the third quantile.

tara_nog_q75 <- quantile(x = tara_weights, 0.75)
tara_nog_q75

##          75% 
## 8.505716e-05

Number of selected edges with the third quantile as threshold:

length(tara_weights[tara_weights >= tara_nog_q75])

## [1] 2398

The following figures show where are these two threshold in the weight distribution.

hist(log(tara_weights, 10), nclass = 100, main = "nog weight distribution", xlab = "log10(weights)")
abline(v = log(tara_nog_median, 10), col = "blue", lwd = 3)
text(log(tara_nog_median, 10), 350, pos = 4, "Median", col = "blue", cex = 1)
abline(v = log(tara_nog_q75, 10), col = "purple", lwd = 3)
text(log(tara_nog_q75, 10), 250, pos = 4, "quantile 75%", col = "purple", cex = 1)

10.4.2.3 Topology network

To determine the range of the threshold, we check the weights.

summary(tara_weights)

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 2.822e-06 1.339e-05 2.877e-05 1.190e-04 8.506e-05 1.208e-02

We define the threshold range to try.

thresholds <- seq(0, 0.008, 0.00005)
length(thresholds)

## [1] 161

Then, we calculate the Average Clustering Coefficient for each threshold.

tara_nog_ACC <- do.call(rbind, lapply(thresholds, function(t, graph){
  graph_sub <- subgraph.edges(graph, E(graph)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(graph_sub), "thresholds" = t, "EN" = length(E(graph_sub)))
  return(df)
}, tara_nog_net))

## ACC
plot(x = tara_nog_ACC$thresholds, y = tara_nog_ACC$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC of orthologous gene data", type = "o")
points(x = tara_nog_ACC$thresholds[1], y = tara_nog_ACC$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = tara_nog_ACC$thresholds[9], y = tara_nog_ACC$ACC[9], col = "pink", pch = 16, cex = 1.2)
points(x = tara_nog_ACC$thresholds[10], y = tara_nog_ACC$ACC[10], col = "purple", pch = 16, cex = 1.2)
abline(v = tara_nog_ACC$thresholds[10], col = "purple")
text(tara_nog_ACC$thresholds[10], 0.8, pos = 4, paste0("Threshold = ",  tara_nog_ACC$thresholds[10]), col = "purple")
text(tara_nog_ACC$thresholds[10], 0.7, pos = 4, paste0("ACCmax = ",  round(tara_nog_ACC$ACC[10], 2)), col = "purple")
## EN
plot(x = tara_nog_ACC$thresholds, y = tara_nog_ACC$EN, xlab = "thresholds", ylab = "number of edges", main = "Edge number of orthologous genes data", type = "o")
abline(v = tara_nog_ACC$thresholds[10], col = "purple")
text(tara_nog_ACC$thresholds[10], 2800, pos = 4, paste0("Threshold = ",  tara_nog_ACC$thresholds[10]), col = "purple")
text(tara_nog_ACC$thresholds[10], 2000, pos = 4, paste0("ACCmax = ",  round(tara_nog_ACC$ACC[10], 2)), col = "purple")
text(tara_nog_ACC$thresholds[10], 1200, pos = 4, paste0("EN = ",  tara_nog_ACC[10, "EN"]), col = "purple")

Figure 10.4: Left: Average Clustering Coeeficient (ACC) values for each threshold - Right: Number of edges on network for each threshold

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.

The local maxima threshold is:

tara_nog_ACC$thresholds[10]

## [1] 0.00045

And the number of selected egdes are:

tara_nog_ACC$EN[10]

## [1] 472

Network visualizations are available in the TARAocean_cytoscape.cys file.

10.4.3 Phylogenetic profil data

tara_phy_net <- graph_from_adjacency_matrix(tara_phy_W, weighted = TRUE, mode = "upper", diag = FALSE)
write.table(as_data_frame(tara_phy_net), "../02_Results/01_TaraOcean/TaraOcean_phy_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

10.4.3.1 Arbitrary threshold

tara_weights <- edge.attributes(tara_phy_net)$weight
hist(tara_phy_W, nclass = 100, main = "phy similarity network weight distribution", xlab = "weights")
hist(log(tara_weights, 10), nclass = 100, main = "phy similarity network weight distribution", xlab = "log10(weights)")
abline(v = log(3.162278e-05, 10), col = "cyan", lwd = 3)

Number of connections:

length(tara_weights[tara_weights>= 3.162278e-05])

## [1] 5188

10.4.3.2 Mean and third quantile

Calculate the median.

tara_phy_median <- median(x = tara_weights)
tara_phy_median

## [1] 3.538224e-05

Number of selected edges with the median as threshold.

length(tara_weights[tara_weights>= tara_phy_median])

## [1] 4796

Calculate the third quantile.

tara_phy_q75 <- quantile(x = tara_weights, 0.75)
tara_phy_q75

##          75% 
## 7.762902e-05

Number of selected edges with the third quantile as threshold.

length(tara_weights[tara_weights>= tara_phy_q75])

## [1] 2398

The following figure show where are these two threshold in the weight distribution.

hist(log(tara_weights, 10), nclass = 100, main = "phy similarity network weight distribution", xlab = "log10(weights)")
abline(v = log(tara_phy_median, 10), col = "blue", lwd = 3)
text(log(tara_phy_median, 10), 370, pos = 4, "Median", col = "blue", cex = 1)
abline(v = log(tara_phy_q75, 10), col = "purple", lwd = 3)
text(log(tara_phy_q75, 10), 250, pos = 4, "quantile 75%", col = "purple", cex = 1)

10.4.3.3 Topology network

We define the threshold range to try.

thresholds <- seq(0, 0.008, 0.00005)
length(thresholds)

## [1] 161

tara_phy_ACC <- do.call(rbind, lapply(thresholds, function(t, graph){
  graph_sub <- subgraph.edges(graph, E(graph)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(graph_sub), "thresholds" = t, "EN" = length(E(graph_sub)))
  return(df)
}, tara_phy_net))

## ACC
plot(x = tara_phy_ACC$thresholds, y = tara_phy_ACC$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC of phylogenetic profil data", type = "o")
points(x = tara_phy_ACC$thresholds[1], y = tara_phy_ACC$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = tara_phy_ACC$thresholds[17], y = tara_phy_ACC$ACC[17], col = "pink", pch = 16, cex = 1.2)
points(x = tara_phy_ACC$thresholds[22], y = tara_phy_ACC$ACC[22], col = "purple", pch = 16, cex = 1.2)
abline(v = tara_phy_ACC$thresholds[22], col = "purple")
text(tara_phy_ACC$thresholds[22], 0.9, pos = 4, paste0("Threshold = ",  tara_phy_ACC$thresholds[22]), col = "purple")
text(tara_phy_ACC$thresholds[22], 0.85, pos = 4, paste0("ACCmax = ",  round(tara_phy_ACC$ACC[22], 2)), col = "purple")
## EN
plot(x = tara_phy_ACC$thresholds, y = tara_phy_ACC$EN, xlab = "thresholds", ylab = "number of edges", main = "Edge number of phylogenetic profil data", type = "o")
abline(v = tara_phy_ACC$thresholds[22], col = "purple")
text(tara_phy_ACC$thresholds[22], 2800, pos = 4, paste0("Threshold = ",  tara_phy_ACC$thresholds[22]), col = "purple")
text(tara_phy_ACC$thresholds[22], 2000, pos = 4, paste0("ACCmax = ",  round(tara_phy_ACC$ACC[22], 2)), col = "purple")
text(tara_phy_ACC$thresholds[22], 1200, pos = 4, paste0("EN = ",  tara_phy_ACC[22, "EN"]), col = "purple")

Figure 10.5: Left: Average Clustering Coeeficient (ACC) values for each threshold - Right: Number of edges on network for each threshold

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.

The local maxima threshold is:

tara_phy_ACC$thresholds[22]

## [1] 0.00105

And the number of selected egdes are:

tara_phy_ACC$EN[22]

## [1] 120

Network visualizations are available in the TARAocean_cytoscape.cys file.

10.5 Downstream analysis

10.5.1 Clustering

Samples are clustered together according to their similarity. According to our data and the information we have, we choose 4 and 8 clusters. Indeed, data are coming from four different depths and eight different oceans.

10.5.1.1 With 4 clusters

C <- 4 
group <- data.frame(Groups = spectralClustering(tara_W, C)) 
row.names(group) <- colnames(tara_W) 
tara_dataGroups4 <- merge(tara_metadata, group, by = 0)

10.5.1.2 With 8 clusters

C <- 8
group <- data.frame(Groups = spectralClustering(tara_W, C)) 
row.names(group) <- colnames(tara_W) 
tara_dataGroups8 <- merge(tara_metadata, group, by = 0)

10.5.1.3 Save results

Results are saved into the same file.

clusters <- merge(x = tara_dataGroups4, y = tara_dataGroups8[c(1,4)], by = "Row.names", suffixes = c("_4clusters", "_8clusters"))
write.table(clusters, "../02_Results/01_TaraOcean/TaraOcean_clusters.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

10.5.2 Visualization with Cytoscape

These are two examples of network visualization for the Tara ocean dataset.

Figure 10.6: Left network: edge weights > 0.014 and node color according ocean. Right network: edge weights > 0.014 and node color according the depth.

Node color represents:
- the ocean (left network)
- the depth (right network)
Node label are the sample names.
Edge color represents the data type contribution for each edge.

We can see a high connected subnetwork on the left, connected to a sparser subnetwork on the right. The highly connected subnetwork corresponds to the MES samples (deeper layer) and the other subnetwork to DCM and SRF (surface layers).

Samples from the same ocean seem to be grouped together. We can’t see this stratification if we analysis one type of data alone.

11 Breast cancer dataset

The breast cancer dataset from The Cancer Genome Atlas (TCGA) contains 3 data types:

mRNA: mRNA expression level
miRNA: microRNA expression level
protein: protein abundance

Data are available in the R package mixOmics. The metadata are also available in this package.

11.1 Input data

11.1.1 Load dataset

We load the breast cancer dataset:

data(breast.TCGA)

The breast.TCGA object contains 3 types of data and one metadata:

names(breast.TCGA$data.train)

## [1] "mirna"   "mrna"    "protein" "subtype"

Dimensions of the data are different:

lapply(breast.TCGA$data.train, dim)

## $mirna
## [1] 150 184
## 
## $mrna
## [1] 150 200
## 
## $protein
## [1] 150 142
## 
## $subtype
## NULL

Data are extracted into single data frame:

tcga_mirna = breast.TCGA$data.train$mirna
tcga_mrna = breast.TCGA$data.train$mrna
tcga_prot = breast.TCGA$data.train$protein

the tcga_miRNA data contain 150 samples in rows and 184 features in columns.
the tcga_mRNA data contain 150 samples in rows and 200 features in columns.
the tcga_prot data contain 150 samples in rows and 142 features in columns.

Data are already well shaped.

tcga_mrna[c(1:5), c(1:5)]

##          RTN2    NDRG2  CCDC113   FAM63A    ACADS
## A0FJ 4.362183 7.533461 3.956124 4.457170 2.256817
## A13E 1.984492 7.455194 5.427623 5.440957 4.028813
## A0G0 1.727323 8.079968 2.227300 5.543480 2.629855
## A0SX 4.363996 5.793750 3.544866 4.737114 4.269101
## A143 2.447562 7.158993 4.691256 4.808728 2.442135

11.1.2 Load metadata

We extract metadat from the breast.TCGA$data.train object.

tcga_metadata = breast.TCGA$data.train$subtype

The metadata contain the subtype of the breast cancer for each sample.

head(tcga_metadata)

## [1] Basal Basal Basal Basal Basal Basal
## Levels: Basal Her2 LumA

For each subtype, there are 45, 30 and 75 samples:

summary(tcga_metadata)

## Basal  Her2  LumA 
##    45    30    75

Metadata should be stored in a data frame:

tcga_metadata_df <- data.frame("subtype" = tcga_metadata)
row.names(tcga_metadata_df) <- row.names(tcga_mirna)

We save the metadata into a file. This file will be useful for the visualization.

write.table(tcga_metadata_df, "../02_Results/03_BreastTCGA/TCGA_metadata.txt", quote = FALSE, row.names = TRUE, col.names = NA, sep = "\t")

11.1.3 Missing data

Data don’t contain missing value. We can go to the following steps.

table(is.na(tcga_mirna))

## 
## FALSE 
## 27600

table(is.na(tcga_mrna))

## 
## FALSE 
## 30000

table(is.na(tcga_prot))

## 
## FALSE 
## 21300

11.1.4 Scaling

We assume that data have been already prepared and normalized.

11.1.4.1 miRNA data

miRNA data are scaled: each column will scaled to have the mean equals to zero and the standard deviation equals to one.

tcga_mirna_scaled <- standardNormalization(x = tcga_mirna)

The following figures are the distribution of the data, before (left) and after (right) scaling. We expected a normal distribution of the data after scaling.

hist(tcga_mirna, nclass = 100, main = "TCGA miRNA data - Data distribution before scaling", xlab = "values")
hist(tcga_mirna_scaled, nclass = 100, main = "TCGA miRNA data - Data distribution after scaling", xlab = "scaled values")

After scaling, data values seem to follow a normal distribution. Data values are centered to zero.

11.1.4.2 mRNA data

mrna data are scaled:

tcga_mrna_scaled <- standardNormalization(x = tcga_mrna)

The following figures are the distribution of the data, before (left) and after (right) scaling. We expected a normal distribution of the data after scaling.

hist(tcga_mrna, nclass = 100, main = "TCGA mRNA data - Data distribution before scaling", xlab = "values")
hist(tcga_mrna_scaled, nclass = 100, main = "TCGA mRNA data - Data distribution after scaling", xlab = "scaled values")

After scaling, data values seem to follow a normal distribution. Data values are centered to zero.

11.1.4.3 Protein data

Protein data are scaled:

tcga_prot_scaled <- standardNormalization(x = tcga_prot)

The following figures are the distribution of the data, before (left) and after (right) scaling. We expected a normal distribution of the data after scaling.

hist(tcga_prot, nclass = 100, main = "TCGA proteomic data - Data distribution before scaling", xlab = "values")
hist(tcga_prot_scaled, nclass = 100, main = "TCGA proteomic data - Data distribution after scaling", xlab = "scaled values")

The protein data seem to be already scaled. So for the following steps, we will used tcga_prot variable.

11.2 Similarity network

In this part, we create the similarity network for each data type.

11.2.1 Distance calculation

We calculate the Euclidean distance between each pair of samples for each type of data.

tcga_mirna_dist <- dist2(tcga_mirna_scaled, tcga_mirna_scaled)
tcga_mrna_dist <- dist2(tcga_mrna_scaled, tcga_mrna_scaled)
tcga_prot_dist <- dist2(tcga_prot, tcga_prot)

Distance matrices have 150 rows and 150 columns. We calculated pairwise distance, so the matrix has samples in rows and in columns.

dim(tcga_mirna_dist)

## [1] 150 150

The diagonal of the distance matrix contains the distance between sample and itself. So the distance is equal (or very close) to zero.

tcga_mirna_dist[c(1:5), c(1:5)]

##          A0FJ         A13E     A0G0     A0SX     A143
## A0FJ   0.0000 3.150041e+02 271.9832 203.1437 513.4011
## A13E 315.0041 5.684342e-14 391.9542 291.6305 421.5542
## A0G0 271.9832 3.919542e+02   0.0000 243.1119 344.3791
## A0SX 203.1437 2.916305e+02 243.1119   0.0000 413.7339
## A143 513.4011 4.215542e+02 344.3791 413.7339   0.0000

High distance values mean that samples are not similar. And small distance values mean that samples are similar.

11.2.2 Similarity calculation

The distance matrix is then transformed into similarity matrix for each data type. We set two parameters:

K = 20: number of nearest neighbors
signma = 0.5: hyperparameter

K <- 20
sigma <- 0.5

The affinityMatrix() function transforms the distance into similarity according the distance with the nearest neighbors.

tcga_mirna_W <- affinityMatrix(tcga_mirna_dist, K, sigma)
tcga_mrna_W <- affinityMatrix(tcga_mrna_dist, K, sigma)
tcga_prot_W <- affinityMatrix(tcga_prot_dist, K, sigma)

The following figures are the heatmap of the similarity matrix (W) of each data type. Samples are clustered using hierarchical clustering. For a better visualization, we log-transform similarities.

pheatmap(log(tcga_mirna_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = tcga_metadata_df, main = "TCGA miRNA data")
pheatmap(log(tcga_mrna_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = tcga_metadata_df, main = "TCGA mRNA data")
pheatmap(log(tcga_prot_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = tcga_metadata_df, main = "TCGA proteomic data")

TCGA dataset - log-transformed similarity matrix heatmap

Figure 11.1: TCGA dataset - log-transformed similarity matrix heatmap

Red color means a high similarity value between two samples whereas blue color means a small similarity value between two samples.

Heatmaps are different between data types.

for miRNA data (top left), there are several groups of samples. Two of them are very different (basal vs lumA)
for mRNA data (top right), we can see two different groups (basal vs lumA)
for protein data (bottom), samples are grouped by their cancer subtypes.

11.3 Fusion

11.3.1 Create the fused similarity matrix

We create the fused similarity matrix using these three parameters:

list that contains the miRNA, mRNA and pro similarity matrices
K = 20: number of nearest neighbors
T = 10: number of iterations

K = 20
T = 10
tcga_W <- SNF(list(tcga_mirna_W, tcga_mrna_W, tcga_prot_W), K, T)
tcga_W[c(1:5), c(1:5)]

##             A0FJ        A13E        A0G0        A0SX        A143
## A0FJ 0.500000000 0.014525731 0.015552881 0.013856749 0.006897882
## A13E 0.014525731 0.500000000 0.015887769 0.008857892 0.006463031
## A0G0 0.015552881 0.015887769 0.500000000 0.003744441 0.011143993
## A0SX 0.013856749 0.008857892 0.003744441 0.500000000 0.003083092
## A143 0.006897882 0.006463031 0.011143993 0.003083092 0.500000000

The dimensions of the fused network are 150 rows and 150 columns, such as the previous similarity matrices. The fused similarity matrix contains similarities between samples, we can also called them weights.

The fused similarity matrix contains 22500 weights.

length(tcga_W)

## [1] 22500

The fused similarity matrix doesn’t contain zero.

table(tcga_W == 0)

## 
## FALSE 
## 22500

The following figure is the heatmap of the fused similarity matrix. Samples are automatically clustered with a hierarchical clustering. Weights are log-transformed for a better visualization.

pheatmap(log(tcga_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = tcga_metadata_df, main = "TCGA - Fused similarity matrix W")

Read color means a high similarity between samples. Blue color means a small similarity between samples.

In this heatmap, samples seem to be well clustered, according the cancer subtype. Basal samples are very different from LumA samples.

11.3.2 Visualize the fused similarity network

tcga_W_net <- graph_from_adjacency_matrix(tcga_W, weighted = TRUE, mode = "upper", diag = FALSE)

Then, the fused similarity network is saved into a the TCGA_W_edgeList.txt file:

write.table(as_data_frame(tcga_W_net), "../02_Results/03_BreastTCGA/TCGA_W_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

This files is loaded into Cytoscape. The Figure 11.2 shows the fused similarity network of the Tara Ocean dataset.

Figure 11.2: Fused similarity network of the breast cancer dataset. Visualization using Cytoscape.

According Cytoscape, the fused similarity network contains 150 nodes (samples) and 11175 edges (connections) between samples. The connections number is smaller in Cytoscape. Indeed, in the similarity matrix weights are duplicates. The similarity matrix contains also the weights for each sample compare to itself (self loops).

For now, the fused similarity network is fully connected: each sample is connected to every other samples. Connections between samples are weighted: some connections are strong (samples are similar) and some other are weak (samples are not similar).

11.4 Threshold selection

In this section, we will determine a threshold to select the strongest connections between samples.

11.4.1 Fused similarity network

11.4.1.1 Arbitrary threshold

tcga_weights <- edge.attributes(tcga_W_net)$weight
hist(tcga_weights, nclass = 100, main = "Fused network weight distribution", xlab = "weights")
hist(log(tcga_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "weights")
abline(v = log(0.0039, 10), col = "cyan", lwd = 3)

The log-transformed weight distribution shows a binomial distribution. We would probably like to cut between the two peaks and choose the corresponding weight: 0.0039. With this threshold, we select 3198 connections.

11.4.1.2 Mean and third quantile

Calculate the median of the weights:

tcga_W_median <- median(x = tcga_weights)
tcga_W_median

## [1] 0.001811571

With the mean (0.0018116) as threshold, we select 5588 connections.

length(tcga_weights[tcga_weights >= tcga_W_median])

## [1] 5588

Calculate the third quantile of the weights:

tcga_W_q75 <- quantile(x = tcga_weights, 0.75)
tcga_W_q75

##         75% 
## 0.004639675

With the third quantile (0.0046397) as threshold, we select 2794 connections.

length(tcga_weights[tcga_weights >= tcga_W_q75])

## [1] 2794

The following figure displays the log-transformed weight distribution with the two previous calculated metrics as markers.

hist(log(tcga_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "log10(weights)")
abline(v = log(tcga_W_median, 10), col = "blue", lwd = 3)
text(log(tcga_W_median, 10), 160, pos = 2, "Median", col = "blue", cex = 1)
abline(v = log(tcga_W_q75, 10), col = "purple", lwd = 3)
text(log(tcga_W_q75, 10), 160, pos = 4, "quantile 75%", col = "purple", cex = 1)

11.4.1.3 Topology network

To determine a range of thresholds to try, we check the weights.

summary(c(tcga_W))

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 0.0001333 0.0006946 0.0018320 0.0066667 0.0047588 0.5000000

We define a vector of threshold range to try (at least 100 values).

thresholds <- seq(0, 0.02, 0.0002)
length(thresholds)

## [1] 101

Then, we calculate the Average Clustering Coefficient for each threshold.

tcga_ACC_W <- do.call(rbind, lapply(thresholds, function(t, graph){
  graph_sub <- subgraph.edges(graph, E(graph)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(graph_sub), "thresholds" = t, "EN" = length(E(graph_sub)))
  return(df)
}, tcga_W_net))

Calculated values are displayed in the following figures.

## ACC
plot(x = tcga_ACC_W$thresholds, y = tcga_ACC_W$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC calculation of the Fused network W", type = "o")
points(x = tcga_ACC_W$thresholds[1], y = tcga_ACC_W$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = tcga_ACC_W$thresholds[55], y = tcga_ACC_W$ACC[55], col = "pink", pch = 16, cex = 1.2)
points(x = tcga_ACC_W$thresholds[58], y = tcga_ACC_W$ACC[58], col = "purple", pch = 16, cex = 1.2)
text(tcga_ACC_W$thresholds[58], 0.5, pos = 4, paste0("Threshold = ",  tcga_ACC_W$thresholds[58]), col = "purple")
text(tcga_ACC_W$thresholds[58], 0.4, pos = 4, paste0("ACCmax = ",  tcga_ACC_W$ACC[58]), col = "purple")
## EN
plot(x = tcga_ACC_W$thresholds, y = tcga_ACC_W$EN, xlab = "thresholds", ylab = "number of edges", main = "EN of the Fused network W", type = "o")
abline(v = tcga_ACC_W$thresholds[58], col = "purple")
text(tcga_ACC_W$thresholds[58], 6000, pos = 4, paste0("Threshold = ",  tcga_ACC_W$thresholds[58]), col = "purple")
text(tcga_ACC_W$thresholds[58], 5000, pos = 4, paste0("ACCmax = ",  tcga_ACC_W$ACC[58]), col = "purple")
text(tcga_ACC_W$thresholds[58], 4000, pos = 4, paste0("EN = ",  tcga_ACC_W[58, "EN"]), col = "purple")

Figure 11.3: Left: Average Clustering Coeeficient (ACC) values for each threshold - Right: Number of edges on network for each threshold

We don’t have obvious and clear local maxima with this dataset.

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.

If we selected the purple local maxima, we will have 598 edges. It could be not enough edges. Let’s see during the visualization.

tcga_ACC_W$thresholds[58]

## [1] 0.0114

11.4.1.4 Visualization using Cytoscape

The network visualization on the left was created with the third quantile (0.0046397). The network visualization on the right was created with the ACC method (0.0114).

The left network contains lot of edges and it’s difficult to see clear connection between samples. Nevertheless, we can see that LumA samples are not connected (very few edges) to the Basal samples.

This trend is also shows in the right network. Samples from the same cancer subtypes are connected together.

The two representations could be interesting. Network visualizations are available in the TCGA_cytoscape.cys file.

11.4.2 miRNA data

tcga_mirna_net <- graph_from_adjacency_matrix(tcga_mirna_W, weighted = TRUE, mode = "upper", diag = FALSE)
write.table(as_data_frame(tcga_mirna_net), "../02_Results/03_BreastTCGA/TCGA_miRNA_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

11.4.2.1 Arbitrary threshold

tcga_weights <- edge.attributes(tcga_mirna_net)$weight
hist(tcga_weights, nclass = 100, main = "Fused network weight distribution", xlab = "weights")
hist(log(tcga_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "weights")
abline(v = log(0.0001, 10), col = "cyan", lwd = 3)

The log-transformed weight distribution shows a kind of normal distribution. We would probably like to cut in the middle of the peak, or just before or after. If we cut in the middle, the corresponding weight is: 0.0001. With this threshold, we select 5011 connections.

11.4.2.2 Mean and third quantile

Calculate the median of the weights:

tcga_mirna_median <- median(x = tcga_weights)
tcga_mirna_median

## [1] 8.421742e-05

Number of selected edges with the median as threshold.

length(tcga_weights[tcga_weights >= tcga_mirna_median])

## [1] 5588

Calculate the third quantile of the weights:

tcga_mirna_q75 <- quantile(x = tcga_weights, 0.75)
tcga_mirna_q75

##          75% 
## 0.0001979659

Number of selected edges with the third quantile as threshold:

length(tcga_weights[tcga_weights >= tcga_mirna_q75])

## [1] 2794

The following figure displays the log-transformed weight distribution with the two previous calculated metrics as markers.

hist(log(tcga_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "log10(weights)")
abline(v = log(tcga_mirna_median, 10), col = "blue", lwd = 3)
text(log(tcga_mirna_median, 10), 400, pos = 2, "Median", col = "blue", cex = 1)
abline(v = log(tcga_mirna_q75, 10), col = "purple", lwd = 3)
text(log(tcga_mirna_q75, 10), 400, pos = 4, "quantile 75%", col = "purple", cex = 1)

11.4.2.3 Topology network

To determine a range of thresholds to try, we check the weights.

summary(c(tcga_mirna_W))

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 3.300e-07 3.072e-05 8.514e-05 2.002e-04 2.020e-04 1.077e-02

We define a vector of threshold range to try (at least 100 values).

thresholds <- seq(0, 0.002, 0.00002)
length(thresholds)

## [1] 101

Then, we calculate the Average Clustering Coefficient for each threshold.

tcga_ACC_mirna <- do.call(rbind, lapply(thresholds, function(t, graph){
  graph_sub <- subgraph.edges(graph, E(graph)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(graph_sub), "thresholds" = t, "EN" = length(E(graph_sub)))
  return(df)
}, tcga_mirna_net))

Calculated values are displayed in the following figures.

## ACC
plot(x = tcga_ACC_mirna$thresholds, y = tcga_ACC_mirna$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC calculation of the Fused network W", type = "o")
points(x = tcga_ACC_mirna$thresholds[1], y = tcga_ACC_mirna$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = tcga_ACC_mirna$thresholds[38], y = tcga_ACC_mirna$ACC[38], col = "pink", pch = 16, cex = 1.2)
points(x = tcga_ACC_mirna$thresholds[41], y = tcga_ACC_mirna$ACC[41], col = "purple", pch = 16, cex = 1.2)
text(tcga_ACC_mirna$thresholds[41], 0.5, pos = 4, paste0("Threshold = ",  tcga_ACC_mirna$thresholds[41]), col = "purple")
text(tcga_ACC_mirna$thresholds[41], 0.4, pos = 4, paste0("ACCmax = ",  tcga_ACC_mirna$ACC[41]), col = "purple")
## EN
plot(x = tcga_ACC_mirna$thresholds, y = tcga_ACC_mirna$EN, xlab = "thresholds", ylab = "number of edges", main = "EN of the Fused network W", type = "o")
abline(v = tcga_ACC_mirna$thresholds[41], col = "purple")
text(tcga_ACC_mirna$thresholds[41], 6000, pos = 4, paste0("Threshold = ",  tcga_ACC_mirna$thresholds[41]), col = "purple")
text(tcga_ACC_mirna$thresholds[41], 5000, pos = 4, paste0("ACCmax = ",  tcga_ACC_mirna$ACC[41]), col = "purple")
text(tcga_ACC_mirna$thresholds[41], 4500, pos = 4, paste0("EN = ",  tcga_ACC_mirna[41, "EN"]), col = "purple")

Figure 11.4: Left: Average Clustering Coeeficient (ACC) values for each threshold - Right: Number of edges on network for each threshold

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.

The local maxima threshold is:

tcga_ACC_mirna$thresholds[41]

## [1] 8e-04

And the number of selected egdes are:

tcga_ACC_mirna$EN[41]

## [1] 249

Network visualizations are available in the TCGA_cytoscape.cys file.

11.4.3 mRNA data

tcga_mrna_net <- graph_from_adjacency_matrix(tcga_mrna_W, weighted = TRUE, mode = "upper", diag = FALSE)
write.table(as_data_frame(tcga_mrna_net), "../02_Results/03_BreastTCGA/TCGA_mRNA_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

11.4.3.1 Arbitrary threshold

tcga_weights <- edge.attributes(tcga_mrna_net)$weight
hist(tcga_weights, nclass = 100, main = "Fused network weight distribution", xlab = "weights")
hist(log(tcga_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "weights")
abline(v = log(0.0001, 10), col = "cyan", lwd = 3)

The log-transformed weight distribution shows a binomial distribution. We would probably like to cut between the two peaks and choose the corresponding weight: 0.0001. With this threshold, we select 4702 connections.

11.4.3.2 Mean and third quantile

Calculate the median of the weights:

tcga_mrna_median <- median(x = tcga_weights)
tcga_mrna_median

## [1] 7.506899e-05

Number of selected edges with the median as threshold.

length(tcga_weights[tcga_weights >= tcga_mrna_median])

## [1] 5588

Calculate the third quantile of the weights:

tcga_mrna_q75 <- quantile(x = tcga_weights, 0.75)
tcga_mrna_q75

##          75% 
## 0.0001834747

Number of selected edges with the third quantile as threshold.

length(tcga_weights[tcga_weights >= tcga_mrna_q75])

## [1] 2794

The following figure displays the log-transformed weight distribution with the two previous calculated metrics as markers.

hist(log(tcga_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "log10(weights)")
abline(v = log(tcga_mrna_median, 10), col = "blue", lwd = 3)
text(log(tcga_mrna_median, 10) - 0.2, 380, pos = 2, "Median", col = "blue", cex = 1)
abline(v = log(tcga_mrna_q75, 10), col = "purple", lwd = 3)
text(log(tcga_mrna_q75, 10), 380, pos = 4, "quantile 75%", col = "purple", cex = 1)

11.4.3.3 Topology network

To determine a range of thresholds to try, we check the weights.

summary(c(tcga_mrna))

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
##  0.007847  4.002122  5.172385  5.255571  6.389464 12.980562

We define a vector of threshold range to try (at least 100 values).

thresholds <- seq(0, 0.0025, 0.00002)
length(thresholds)

## [1] 126

Then, we calculate the Average Clustering Coefficient for each threshold.

tcga_ACC_mrna <- do.call(rbind, lapply(thresholds, function(t, graph){
  graph_sub <- subgraph.edges(graph, E(graph)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(graph_sub), "thresholds" = t, "EN" = length(E(graph_sub)))
  return(df)
}, tcga_mrna_net))

Calculated values are displayed in the following figures.

## ACC
plot(x = tcga_ACC_mrna$thresholds, y = tcga_ACC_mrna$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC calculation of the Fused network W", type = "o")
points(x = tcga_ACC_mrna$thresholds[1], y = tcga_ACC_mrna$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = tcga_ACC_mrna$thresholds[54], y = tcga_ACC_mrna$ACC[54], col = "pink", pch = 16, cex = 1.2)
points(x = tcga_ACC_mrna$thresholds[56], y = tcga_ACC_mrna$ACC[56], col = "purple", pch = 16, cex = 1.2)
text(tcga_ACC_mrna$thresholds[56], 0.5, pos = 4, paste0("Threshold = ",  tcga_ACC_mrna$thresholds[56]), col = "purple")
text(tcga_ACC_mrna$thresholds[56], 0.4, pos = 4, paste0("ACCmax = ",  tcga_ACC_mrna$ACC[56]), col = "purple")
## EN
plot(x = tcga_ACC_mrna$thresholds, y = tcga_ACC_mrna$EN, xlab = "thresholds", ylab = "number of edges", main = "EN of the Fused network W", type = "o")
abline(v = tcga_ACC_mrna$thresholds[56], col = "purple")
text(tcga_ACC_mrna$thresholds[56], 6000, pos = 4, paste0("Threshold = ",  tcga_ACC_mrna$thresholds[56]), col = "purple")
text(tcga_ACC_mrna$thresholds[56], 5000, pos = 4, paste0("ACCmax = ",  tcga_ACC_mrna$ACC[56]), col = "purple")
text(tcga_ACC_mrna$thresholds[56], 4500, pos = 4, paste0("EN = ",  tcga_ACC_mrna[56, "EN"]), col = "purple")

Figure 11.5: Left: Average Clustering Coeeficient (ACC) values for each threshold - Right: Number of edges on network for each threshold

We don’t have obvious and clear local maxima with this dataset.

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.

The local maxima threshold is:

tcga_ACC_mrna$thresholds[56]

## [1] 0.0011

And the number of selected egdes are:

tcga_ACC_mrna$EN[56]

## [1] 68

Network visualizations are available in the TCGA_cytoscape.cys file.

11.4.4 Protein data

tcga_prot_net <- graph_from_adjacency_matrix(tcga_prot_W, weighted = TRUE, mode = "upper", diag = FALSE)
write.table(as_data_frame(tcga_prot_net), "../02_Results/03_BreastTCGA/TCGA_prot_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

11.4.4.1 Arbitrary threshold

tcga_weights <- edge.attributes(tcga_prot_net)$weight
hist(tcga_weights, nclass = 100, main = "Fused network weight distribution", xlab = "weights")
hist(log(tcga_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "weights")
abline(v = log(7.943282e-05, 10), col = "cyan", lwd = 3)

The log-transformed weight distribution shows a binomial distribution. We would probably like to cut between the two peaks and choose the corresponding weight: 7.943282e-05. With this threshold, we select 6814 connections.

11.4.4.2 Mean and third quantile

Calculate the median of the weights:

tcga_prot_median <- median(x = tcga_weights)
tcga_prot_median

## [1] 0.0001414766

Number of selected edges with the median as threshold.

length(tcga_weights[tcga_weights >= tcga_prot_median])

## [1] 5588

Calculate the third quantile of the weights:

tcga_prot_q75 <- quantile(x = tcga_weights, 0.75)
tcga_prot_q75

##          75% 
## 0.0005063203

Number of selected edges with the third quantile as threshold.

length(tcga_weights[tcga_weights >= tcga_prot_q75])

## [1] 2794

The following figure displays the log-transformed weight distribution with the two previous calculated metrics as markers.

hist(log(tcga_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "log10(weights)")
abline(v = log(tcga_prot_median, 10), col = "blue", lwd = 3)
text(log(tcga_prot_median, 10), 280, pos = 2, "Median", col = "blue", cex = 1)
abline(v = log(tcga_prot_q75, 10), col = "purple", lwd = 3)
text(log(tcga_prot_q75, 10), 260, pos = 4, "quantile 75%", col = "purple", cex = 1)

11.4.4.3 Topology network

To determine a range of thresholds to try, we check the weights.

summary(c(tcga_prot))

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -5.98579 -0.22688  0.00000  0.03095  0.26711  6.63490

We define a vector of threshold range to try (at least 100 values).

thresholds <- seq(0, 0.01, 0.0001)
length(thresholds)

## [1] 101

Then, we calculate the Average Clustering Coefficient for each threshold.

tcga_ACC_prot <- do.call(rbind, lapply(thresholds, function(t, graph){
  graph_sub <- subgraph.edges(graph, E(graph)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(graph_sub), "thresholds" = t, "EN" = length(E(graph_sub)))
  return(df)
}, tcga_prot_net))

Calculated values are displayed in the following figures.

## ACC
plot(x = tcga_ACC_prot$thresholds, y = tcga_ACC_prot$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC calculation of the Fused network W", type = "o")
points(x = tcga_ACC_prot$thresholds[1], y = tcga_ACC_prot$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = tcga_ACC_prot$thresholds[36], y = tcga_ACC_prot$ACC[36], col = "pink", pch = 16, cex = 1.2)
points(x = tcga_ACC_prot$thresholds[39], y = tcga_ACC_prot$ACC[39], col = "purple", pch = 16, cex = 1.2)
text(tcga_ACC_prot$thresholds[39], 0.5, pos = 4, paste0("Threshold = ",  tcga_ACC_prot$thresholds[39]), col = "purple")
text(tcga_ACC_prot$thresholds[39], 0.4, pos = 4, paste0("ACCmax = ",  tcga_ACC_prot$ACC[39]), col = "purple")
## EN
plot(x = tcga_ACC_prot$thresholds, y = tcga_ACC_prot$EN, xlab = "thresholds", ylab = "number of edges", main = "EN of the Fused network W", type = "o")
abline(v = tcga_ACC_prot$thresholds[39], col = "purple")
text(tcga_ACC_prot$thresholds[39], 6000, pos = 4, paste0("ACCmax = ",  tcga_ACC_prot$thresholds[39]), col = "purple")
text(tcga_ACC_prot$thresholds[39], 5000, pos = 4, paste0("ACCmax = ",  tcga_ACC_prot$ACC[39]), col = "purple")
text(tcga_ACC_prot$thresholds[39], 4500, pos = 4, paste0("EN = ",  tcga_ACC_prot[39, "EN"]), col = "purple")

Figure 11.6: Left: Average Clustering Coeeficient (ACC) values for each threshold - Right: Number of edges on network for each threshold

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.

The local maxima threshold is:

tcga_ACC_prot$thresholds[39]

## [1] 0.0038

And the number of selected egdes are:

tcga_ACC_prot$EN[39]

## [1] 186

Network visualizations are available in the TCGA_cytoscape.cys file.

11.5 Downstream analysis

11.5.1 Clustering

Samples are clustered together according to their similarity.We know that there are three breast cancer subtypes in the dataset. So we decide to perform a clustering with three and four clusters.

11.5.1.1 With 3 clusters

C <- 3
group <- data.frame(Groups = spectralClustering(tcga_W, C)) 
row.names(group) <- colnames(tcga_W) 
tcga_dataGroups3 <- merge(tcga_metadata_df, group, by = 0)

11.5.1.2 With 4 clusters

C <- 4
group <- data.frame(Groups = spectralClustering(tcga_W, C)) 
row.names(group) <- colnames(tcga_W) 
tcga_dataGroups4 <- merge(tcga_metadata_df, group, by = 0)

11.5.1.3 Save results

Results are saved into the same file.

clusters <- merge(x = tcga_dataGroups3, y = tcga_dataGroups4[-2], by = "Row.names", suffixes = c("_3clusters", "_4clusters"))
write.table(clusters, "../02_Results/03_BreastTCGA/TCGA_clusters.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

11.5.2 Visualization with Cytoscape

These are two examples of network visualization for the breast cancer dataset.

**Left network**: edge weights > 0.011 and node color according cancer subtypes. **Right network**: edge weights > 0.011 and node color according clustering results.

Figure 11.7: Left network: edge weights > 0.011 and node color according cancer subtypes. Right network: edge weights > 0.011 and node color according clustering results.

Node color represents
- the cancer subtypes (left network)
- the clustering results (right network), we selected three clusters.
Node label are the sample names.
Edge color represents the data type contribution for each edge.

We can see three interconnected groups. These groups are consistent with the cancer subtypes. We can also assign one cancer subtype per clusters, found by SNF.

Protein data support a lot of edges in this network. And miRNA and mRNA data seem to capture same kind of information (light green edges).

12 CLL dataset

The Chronic Lymphocytic Leukaemia (CLL) dataset contains 4 data types:

mRNA: transcriptom expression level
methylation: DNA methylation assays
drug: drug response measurements
mutation: sommatic mutation status

Data are available in the R package MOFAdata. Metadata file is available in the summer school’s GitHub repository.

12.1 Input data

12.1.1 Load dataset

The CLL data are available in the MOFAdata R package.

data("CLL_data")

The CLL_data object contains 4 types of data with different dimensions:

lapply(CLL_data, dim)

## $Drugs
## [1] 310 200
## 
## $Methylation
## [1] 4248  200
## 
## $mRNA
## [1] 5000  200
## 
## $Mutations
## [1]  69 200

Rows are features (e.g. drug, genes) and columns are samples. There are 200 samples. We have to change the shape of the data.

CLL_data_t <- lapply(CLL_data, t)

Now, samples are in rows:

CLL_data_t$mRNA[c(1:5), c(1:5)]

##      ENSG00000244734 ENSG00000158528 ENSG00000198478 ENSG00000175445
## H045        4.558644       11.741854        8.921456       12.686458
## H109        2.721512       13.287432        2.721512       10.925985
## H024        9.938456        2.341006       12.381452        1.528848
## H056       13.278004        3.232874        8.106266        1.528848
## H079        6.086874       11.940820        4.889503       13.340588
##      ENSG00000174469
## H045        2.644946
## H109       12.648355
## H024        1.528848
## H056       13.565210
## H079        5.476914

12.1.2 Load metadata

The sample_metadata.txt file contains metadata. It contains header (head = TRUE) and row names (row.names = 1).

CLL_metadata <- read.table("../00_Data/CLL/sample_metadata.txt", head = TRUE, sep = "\t", row.names = 1)
head(CLL_metadata)

##      Gender      age        TTT      TTD treatedAfter  died IGHV trisomy12
## H005      m 75.26575 0.57494867 2.625599         TRUE FALSE    1         0
## H006      m       NA         NA       NA           NA    NA   NA        NA
## H007      f       NA         NA       NA           NA    NA   NA        NA
## H008      m       NA         NA       NA           NA    NA   NA        NA
## H010      f 72.78082 2.93223819 2.932238        FALSE FALSE    0         0
## H011      f 72.99452 0.01916496 2.951403         TRUE FALSE    1         0

We have information for each sample about:

names(CLL_metadata)

## [1] "Gender"       "age"          "TTT"          "TTD"          "treatedAfter"
## [6] "died"         "IGHV"         "trisomy12"

For visualization, columns should be numerical, logical or character.

str(CLL_metadata)

## 'data.frame':    200 obs. of  8 variables:
##  $ Gender      : chr  "m" "m" "f" "m" ...
##  $ age         : num  75.3 NA NA NA 72.8 ...
##  $ TTT         : num  0.575 NA NA NA 2.932 ...
##  $ TTD         : num  2.63 NA NA NA 2.93 ...
##  $ treatedAfter: logi  TRUE NA NA NA FALSE TRUE ...
##  $ died        : logi  FALSE NA NA NA FALSE FALSE ...
##  $ IGHV        : int  1 NA NA NA 0 1 0 0 0 0 ...
##  $ trisomy12   : int  0 NA NA NA 0 0 0 0 0 1 ...

CLL_metadata$died <- as.character(CLL_metadata$died)
CLL_metadata$IGHV <- as.character(CLL_metadata$IGHV)
CLL_metadata$trisomy12 <- as.character(CLL_metadata$trisomy12)

12.1.3 Missing data

12.1.3.1 Drug data

Overview of the drug data:

CLL_data_t$Drugs[c(1:5), c(1:5)]

##         D_001_1    D_001_2   D_001_3   D_001_4   D_001_5
## H045 0.02363938 0.04623274 0.3187471 0.8237027 0.8962777
## H109 0.07359900 0.10623002 0.2732891 0.7171379 0.8850003
## H024         NA         NA        NA        NA        NA
## H056 0.05813930 0.09022028 0.2322145 0.7225736 0.7957497
## H079 0.02042077 0.04750543 0.3638962 0.8073907 0.8794886

The CLL drug data contains missing data.

table(is.na(CLL_data_t$Drugs))

## 
## FALSE  TRUE 
## 57040  4960

12.1.3.2 Methylation data

Overview of the methylation data:

CLL_data_t$Methylation[c(1:5), c(1:5)]

##       cg10146935 cg26837773 cg17801765 cg13244315 cg06181703
## H045  1.81108585 -5.1725723  5.4115263 -0.1188251  5.1203838
## H109 -3.99750846  1.5948702  5.4126925  1.0438706  1.2794803
## H024 -2.84431298  0.1611705  0.3657059 -4.2192362  0.7211004
## H056 -3.33865611 -2.0934326  0.3736342 -1.5921965  4.0470594
## H079 -0.01936203  3.7489796  5.4120096  1.4164183  5.2374225

The CLL methylation data contains missing data.

table(is.na(CLL_data_t$Methylation))

## 
##  FALSE   TRUE 
## 832608  16992

12.1.3.3 mRNA data

Overview of the mRNA data:

CLL_data_t$mRNA[c(1:5), c(1:5)]

##      ENSG00000244734 ENSG00000158528 ENSG00000198478 ENSG00000175445
## H045        4.558644       11.741854        8.921456       12.686458
## H109        2.721512       13.287432        2.721512       10.925985
## H024        9.938456        2.341006       12.381452        1.528848
## H056       13.278004        3.232874        8.106266        1.528848
## H079        6.086874       11.940820        4.889503       13.340588
##      ENSG00000174469
## H045        2.644946
## H109       12.648355
## H024        1.528848
## H056       13.565210
## H079        5.476914

The CLL mRNA data contains missing data.

table(is.na(CLL_data_t$mRNA))

## 
##  FALSE   TRUE 
## 680000 320000

12.1.3.4 Mutation data

Overview of the mutation data:

CLL_data_t$Mutations[c(1:5), c(1:5)]

##      gain2p25.3 gain3q26 del6p21.2 del6q21 del8p12
## H045          0        0         0       0       0
## H109          0        0         0       0       0
## H024          0        0         0       0       0
## H056          0        0         0       0       0
## H079          1        0         0       0       0

The CLL mutation data contains missing data.

table(is.na(CLL_data_t$Mutations))

## 
## FALSE  TRUE 
##  9141  4659

12.1.3.5 Remove missing data

We remove samples with at least one missing data in each data type using the NARemoving() function. We set:

margin = 1 because samples are in row
threshold = 0 because we don’t want missing data at all

CLL_drug <- NARemoving(data = CLL_data_t$Drugs, margin = 1, threshold = 0)

## [1] "Remove 16 samples."

CLL_meth <- NARemoving(data = CLL_data_t$Methylation, margin = 1, threshold = 0)

## [1] "Remove 4 samples."

CLL_mrna <- NARemoving(data = CLL_data_t$mRNA, margin = 1, threshold = 0)

## [1] "Remove 64 samples."

CLL_muta <- NARemoving(data = CLL_data_t$Mutations, margin = 1, threshold = 0)

## [1] "Remove 192 samples."

We decide to not use the mutation data because this data contains a lot of missing data. Data types need to have the same set of samples.

sampleNames <- Reduce(intersect, list(rownames(CLL_drug), rownames(CLL_meth), rownames(CLL_mrna)))
CLL_drug <- CLL_drug[rownames(CLL_drug) %in% sampleNames,]
CLL_meth <- CLL_meth[rownames(CLL_meth) %in% sampleNames,]
CLL_mrna <- CLL_mrna[rownames(CLL_mrna) %in% sampleNames,]
lapply(list("Drugs" =  CLL_drug, "Meth" = CLL_meth, "mRNA" = CLL_mrna), dim)

## $Drugs
## [1] 121 310
## 
## $Meth
## [1]  121 4248
## 
## $mRNA
## [1]  121 5000

We will run SNF with 121 samples and three different data types.

12.1.4 Scaling

We assume that data have been already prepared and normalized.

12.1.4.1 Drug data

Drug data are scaled. Each column will have the mean equals to zero and the standard deviation equals to one.

CLL_drug_scaled <- standardNormalization(x = CLL_drug)

The following figures are the distribution of the data, before (left) and after (right) scaling. We expected a normal distribution of the data after scaling.

hist(CLL_drug, nclass = 100, main = "CLL drug data - Data distribution before scaling", xlab = "values")
hist(CLL_drug_scaled, nclass = 100, main = "CLL drug data - Data distribution after scaling", xlab = "scaled values")

After scaling, drug data follow a normal distribution.

12.1.4.2 Methylation data

Methylation data are scaled.

CLL_meth_scaled <- standardNormalization(x = CLL_meth)

The following figures are the distribution of the data, before (left) and after (right) scaling. We expected a normal distribution of the data after scaling.

hist(CLL_meth, nclass = 100, main = "CLL methylation data - Data distribution before scaling", xlab = "values")
hist(CLL_meth_scaled, nclass = 100, main = "CLL methylation data - Data distribution after scaling", xlab = "scaled values")

Here, we can see more a binomial distribution after scaling. But, the data are centered.

12.1.4.3 mRNA data

mRNA data are scaled.

CLL_mrna_scaled <- standardNormalization(x = CLL_mrna)

The following figures are the distribution of the data, before (left) and after (right) scaling. We expected a normal distribution of the data after scaling.

hist(CLL_mrna, nclass = 100, main = "CLL mRNA data - Data distribution before scaling", xlab = "values")
hist(CLL_mrna_scaled, nclass = 100, main = "CLL mRNA data - Data distribution after scaling", xlab = "scaled values")

After scaling, mRNA data follow a normal distribution.

12.2 Similarity network

In this part, we create the similarity network for each data type.

12.2.1 Distance calculation

We calculate the Euclidean distance between each pair of samples for each type of data.

CLL_drug_dist <- dist2(CLL_drug_scaled, CLL_drug_scaled)
CLL_meth_dist <- dist2(CLL_meth_scaled, CLL_meth_scaled)
CLL_mrna_dist <- dist2(CLL_mrna_scaled, CLL_mrna_scaled)

Distance matrices have 121 rows (samples) and 121 columns (samples). We calculated pairwise distance, so the matrix has samples in rows and in columns.

dim(CLL_drug_dist)

## [1] 121 121

The diagonal of the distance matrix contains the distance between sample and itself. So the distance is equal (or very close) to zero.

CLL_drug_dist[c(1:5), c(1:5)]

##              H045         H109         H056     H079     H164
## H045 2.273737e-13 4.028353e+02 1.115350e+03 340.5212 554.7438
## H109 4.028353e+02 2.273737e-13 1.074784e+03 671.3817 608.1489
## H056 1.115350e+03 1.074784e+03 1.136868e-13 729.6333 625.2601
## H079 3.405212e+02 6.713817e+02 7.296333e+02   0.0000 481.3647
## H164 5.547438e+02 6.081489e+02 6.252601e+02 481.3647   0.0000

High distance values mean that samples are not similar. And small distance values mean that samples are similar.

12.2.2 Similarity calculation

The distance values are transformed according the neighbors of the samples. We set two parameters:

K = 20: number of nearest neighbors
sigma = 0.5: hyperparameter

K = 20
sigma = 0.5

The affinityMatrix() function transforms the distance into similarity according the distance with the nearest neighbors.

CLL_drug_W <- affinityMatrix(CLL_drug_dist, K, sigma)
CLL_meth_W <- affinityMatrix(CLL_meth_dist, K, sigma)
CLL_mrna_W <- affinityMatrix(CLL_mrna_dist, K, sigma)

The following figures are the heatmap of the similarity matrix (W) of each data type. Samples are clustered using hierarchical clustering. For a better visualization, we log-transform similarities.

pheatmap(log(CLL_drug_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = CLL_metadata[c(1, 2, 6, 7, 8)], main = "CLL Drugs")
pheatmap(log(CLL_meth_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = CLL_metadata[c(1, 2, 6, 7, 8)], main = "CLL Methylation")
pheatmap(log(CLL_mrna_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = CLL_metadata[c(1, 2, 6, 7, 8)], main = "CLL mRNA")

CLL dataset - log-transformed similarity matrix heatmap

Figure 12.1: CLL dataset - log-transformed similarity matrix heatmap

The red color means high similarity between two samples. The blue color means small similarity between two samples.

Heatmaps are different between data types. Each data seems to carry different information about samples. - drug data: the heatmap shows two groups of similar samples, but non of them seems to be related to a specific metadata. - methylation data: the heatmap shows two or maybe three groups of similar samples. Groups seem to be related to the IGHV status. - mRNA data: the heatmap shows small groups but not clear one.

12.3 Fusion

12.3.1 Create the fused similarity matrix

To create the fused similarity matrix, we set three parameters:

list of similarity matrices (drug, methylation and mRNA)
K = 20: number of nearest neighbors
T = 10: number of iterations

K <- 20
T <- 10
CLL_W <- SNF(list(CLL_drug_W, CLL_meth_W, CLL_mrna_W), K, T)
CLL_W[c(1:5), c(1:5)]

##              H045         H109         H056         H079        H164
## H045 0.5000000000 0.0157070950 0.0008608448 0.0186693619 0.010246238
## H109 0.0157070950 0.5000000000 0.0009468912 0.0085345260 0.004235980
## H056 0.0008608448 0.0009468912 0.5000000000 0.0008622857 0.000796166
## H079 0.0186693619 0.0085345260 0.0008622857 0.5000000000 0.005578270
## H164 0.0102462384 0.0042359797 0.0007961660 0.0055782704 0.500000000

The dimensions of the fused network are 121 rows and 121 columns, such as the previous similarity matrices. The fused similarity matrix contains similarities between samples, we can also called them weights.

dim(CLL_W)

## [1] 121 121

The fused similarity matrix contains 14641 weights.

length(CLL_W)

## [1] 14641

The fused similarity matrix doesn’t contain zero.

length(CLL_W[length(CLL_W) == 0])

## [1] 0

The following figure is the heatmap of the fused similarity matrix. Samples are automatically clustered with a hierarchical clustering. Weights are log-transformed for a better visualization.

pheatmap(log(CLL_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = CLL_metadata[c(1, 2, 6, 7, 8)], main = "CLL - Fused similarity matrix W")

The red color means high similarity between two samples. The blue color means small similarity between two samples.

The heatmap shows two main groups. Samples between groups are very different. Groups seem to be related to the IGHV status. This heatmap doesn’t give us obvious information. We can make several assumptions:

it’s the result
a previous step wasn’t the best for one or several data type (normalization, scaling, distance etc)
parameters used (K, sigma and T) are not the most adapted.

It could be interesting to try another distance and/or try with different parameters.

12.3.2 Visualized the fused similarity network

CLL_W_net <- graph_from_adjacency_matrix(CLL_W, weighted = TRUE, mode = "upper", diag = FALSE)

Then, the fused similarity network is saved into a the CLL_W_edgeList.txt file:

write.table(as_data_frame(CLL_W_net), "../02_Results/02_CLL/CLL_W_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

This files is loaded into Cytoscape. The Figure ?? shows the fused similarity network of the Tara Ocean dataset.

According Cytoscape, the fused similarity network contains 121 nodes (samples) and 7260 edges (connections) between samples. The connections number is smaller in Cytoscape. Indeed, in the similarity matrix weights are duplicates. The similarity matrix contains also the weights for each sample compare to itself (self loops).

12.4 Threshold selection

In this section, we will determine a threshold to select the strongest connections between samples.

12.4.1 Fused similarity network

12.4.1.1 Arbitrary threshold

CLL_weights <- edge.attributes(CLL_W_net)$weight
hist(CLL_weights, nclass = 100, main = "Fused network weight distribution", xlab = "weights")
hist(log(CLL_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "weights")
abline(v = log(0.004466836, 10), col = "cyan", lwd = 3)

The log-transformed weight distribution shows a binomial distribution. We would probably like to cut between the two peaks and choose the corresponding weight: 0.004466836. With this threshold, we select 2457 connections.

12.4.1.2 Mean and third quantile

We calculate the median of the weights.

CLL_W_median <- median(x = CLL_weights)
CLL_W_median

## [1] 0.002700864

With the mean (0.0027009) as threshold, we select 3630 connections.

length(CLL_weights[CLL_weights >= CLL_W_median])

## [1] 3630

Calculate the third quantile of the weights:

CLL_W_q75 <- quantile(x = CLL_weights, 0.75)
CLL_W_q75

##         75% 
## 0.005983555

With the third quantile (0.0059836) as threshold, we select 1815 connections.

length(CLL_weights[CLL_weights >= CLL_W_q75])

## [1] 1815

The following figure displays the log-transformed weight distribution with the two previous calculated metrics as markers.

hist(log(CLL_weights, 10), nclass = 100, main = "Fused similarity network weight distribution", xlab = "log10(weights)")
abline(v = log(CLL_W_median, 10), col = "blue", lwd = 3)
text(log(CLL_W_median, 10), 160, pos = 4, "Median", col = "blue", cex = 1)
abline(v = log(CLL_W_q75, 10), col = "purple", lwd = 3)
text(log(CLL_W_q75, 10), 160, pos = 4, "quantile 75%", col = "purple", cex = 1)

12.4.1.3 Topology network

To determine a range of thresholds to try, we check the weights.

summary(c(CLL_W))

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 0.0004991 0.0014801 0.0027246 0.0082645 0.0061004 0.5000000

We define a vector of threshold range to try (at least 100 values).

thresholds <- seq(0, 0.03, 0.0003)
length(thresholds)

## [1] 101

Then, we calculate the Average Clustering Coefficient for each threshold.

CLL_ACC_W <- do.call(rbind, lapply(thresholds, function(t, graph){
  graph_sub <- subgraph.edges(graph, E(graph)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(graph_sub), "thresholds" = t, "EN" = length(E(graph_sub)))
  return(df)
}, CLL_W_net))

Calculated values are displayed in the following figures.

## ACC
plot(x = CLL_ACC_W$thresholds, y = CLL_ACC_W$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC calculation of the Fused network W", type = "o")
points(x = CLL_ACC_W$thresholds[1], y = CLL_ACC_W$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = CLL_ACC_W$thresholds[41], y = CLL_ACC_W$ACC[41], col = "pink", pch = 16, cex = 1.2)
points(x = CLL_ACC_W$thresholds[42], y = CLL_ACC_W$ACC[42], col = "purple", pch = 16, cex = 1.2)
text(CLL_ACC_W$thresholds[42], 0.5, pos = 4, paste0("Threshold = ",  CLL_ACC_W$thresholds[42]), col = "purple")
text(CLL_ACC_W$thresholds[42], 0.4, pos = 4, paste0("ACCmax = ",  CLL_ACC_W$ACC[42]), col = "purple")
## EN
plot(x = CLL_ACC_W$thresholds, y = CLL_ACC_W$EN, xlab = "thresholds", ylab = "number of edges", main = "EN of the Fused network W", type = "o")
abline(v = CLL_ACC_W$thresholds[42], col = "purple")
text(CLL_ACC_W$thresholds[42], 6000, pos = 4, paste0("Threshold = ",  CLL_ACC_W$thresholds[42]), col = "purple")
text(CLL_ACC_W$thresholds[42], 5000, pos = 4, paste0("ACCmax = ",  CLL_ACC_W$ACC[42]), col = "purple")
text(CLL_ACC_W$thresholds[42], 4500, pos = 4, paste0("EN = ",  CLL_ACC_W[42, "EN"]), col = "purple")

Figure 12.2: Left: Average Clustering Coeeficient (ACC) values for each threshold - Right: Number of edges on network for each threshold

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.

If we selected the purple local maxima, we will have 312 edges. It could be not enough edges. Let’s see during the visualization.

CLL_ACC_W$thresholds[42]

## [1] 0.0123

12.4.1.4 Visualization using Cytoscape

The network visualization on the left was created with the third quantile (0.0059836). The network visualization on the right was created with the ACC method (0.0123).

The left network contains lot of edges and it’s difficult to see clear connection between samples. Nevertheless, we can see two groups of samples, according the IGVH status.

This trend is also shows in the right network. Samples with same IGVH status are connected together.

It could be interesting to map other metadata on the network. Network visualizations are available in the CLL_cytoscape.cys file.

12.4.2 Drug data

CLL_drug_net <- graph_from_adjacency_matrix(CLL_drug_W, weighted = TRUE, mode = "upper", diag = FALSE)
write.table(as_data_frame(CLL_drug_net), "../02_Results/02_CLL/CLL_drug_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

12.4.2.1 Arbitrary threshold

CLL_weights <- edge.attributes(CLL_drug_net)$weight
hist(CLL_weights, nclass = 100, main = "Fused network weight distribution", xlab = "weights")
hist(log(CLL_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "weights")
abline(v = log(0.00007, 10), col = "cyan", lwd = 3)

The log-transformed weight distribution shows a binomial distribution. We would probably like to cut between the two peaks and choose the corresponding weight: 0.00007. With this threshold, we select 3210 connections.

12.4.2.2 Mean and third quantile

We calculate the median of the weights.

CLL_drug_median <- median(x = CLL_weights)
CLL_drug_median

## [1] 5.703429e-05

Number of selected edges with the median as threshold.

length(CLL_weights[CLL_weights >= CLL_drug_median])

## [1] 3630

Calculate the third quantile of the weights:

CLL_drug_q75 <- quantile(x = CLL_weights, 0.75)
CLL_drug_q75

##          75% 
## 0.0001371145

Number of selected edges with the third quantile as threshold:

length(CLL_weights[CLL_weights >= CLL_drug_q75])

## [1] 1815

The following figure displays the log-transformed weight distribution with the two previous calculated metrics as markers.

hist(log(CLL_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "log10(weights)")
abline(v = log(CLL_drug_median, 10), col = "blue", lwd = 3)
text(log(CLL_drug_median, 10) - 0.5, 200, pos = 2, "Median", col = "blue", cex = 1)
abline(v = log(CLL_drug_q75, 10), col = "purple", lwd = 3)
text(log(CLL_drug_q75, 10), 230, pos = 4, "quantile 75%", col = "purple", cex = 1)

12.4.2.3 Topology network

To determine a range of thresholds to try, we check the weights.

summary(c(CLL_drug_W))

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 2.650e-07 2.170e-05 5.803e-05 1.347e-04 1.394e-04 5.758e-03

We define a vector of threshold range to try (at least 100 values).

thresholds <- seq(0, 0.0015, 0.00001)
length(thresholds)

## [1] 151

Then, we calculate the Average Clustering Coefficient for each threshold.

CLL_ACC_drug <- do.call(rbind, lapply(thresholds, function(t, graph){
  graph_sub <- subgraph.edges(graph, E(graph)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(graph_sub), "thresholds" = t, "EN" = length(E(graph_sub)))
  return(df)
}, CLL_drug_net))

Calculated values are displayed in the following figures.

## ACC
plot(x = CLL_ACC_drug$thresholds, y = CLL_ACC_drug$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC calculation of the Fused network W", type = "o")
points(x = CLL_ACC_drug$thresholds[1], y = CLL_ACC_drug$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = CLL_ACC_drug$thresholds[39], y = CLL_ACC_drug$ACC[39], col = "pink", pch = 16, cex = 1.2)
points(x = CLL_ACC_drug$thresholds[45], y = CLL_ACC_drug$ACC[45], col = "purple", pch = 16, cex = 1.2)
text(CLL_ACC_drug$thresholds[45], 0.5, pos = 4, paste0("Threshold = ",  CLL_ACC_drug$thresholds[45]), col = "purple")
text(CLL_ACC_drug$thresholds[45], 0.4, pos = 4, paste0("ACCmax = ",  CLL_ACC_drug$ACC[45]), col = "purple")
## EN
plot(x = CLL_ACC_drug$thresholds, y = CLL_ACC_drug$EN, xlab = "thresholds", ylab = "number of edges", main = "EN of the Fused network W", type = "o")
abline(v = CLL_ACC_drug$thresholds[42], col = "purple")
text(CLL_ACC_drug$thresholds[45], 5300, pos = 4, paste0("Threshold = ",  CLL_ACC_drug$thresholds[45]), col = "purple")
text(CLL_ACC_drug$thresholds[45], 5000, pos = 4, paste0("ACCmax = ",  CLL_ACC_drug$ACC[45]), col = "purple")
text(CLL_ACC_drug$thresholds[45], 4700, pos = 4, paste0("EN = ",  CLL_ACC_drug[45, "EN"]), col = "purple")

Figure 12.3: Left: Average Clustering Coeeficient (ACC) values for each threshold - Right: Number of edges on network for each threshold

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.

If we selected the purple local maxima, we will have 252 edges. It could be not enough edges. Let’s see during the visualization.

CLL_ACC_drug$thresholds[45]

## [1] 0.00044

Network visualizations are available in the CLL_cytoscape.cys file.

12.4.3 Methylation data

CLL_meth_net <- graph_from_adjacency_matrix(CLL_meth_W, weighted = TRUE, mode = "upper", diag = FALSE)
write.table(as_data_frame(CLL_meth_net), "../02_Results/02_CLL/CLL_meth_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

12.4.3.1 Arbitrary threshold

CLL_weights <- edge.attributes(CLL_meth_net)$weight
hist(CLL_weights, nclass = 100, main = "Fused network weight distribution", xlab = "weights")
hist(log(CLL_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "weights")
abline(v = log(1.258925e-05, 10), col = "cyan", lwd = 3)

The log-transformed weight distribution shows a kind of normal distribution. We probably would like to cut the peaks and choose the corresponding weight: 1.258925e-05. With this threshold, we select 1393 connections.

12.4.3.2 Mean and third quantile

We calculate the median of the weights.

CLL_meth_median <- median(x = CLL_weights)
CLL_meth_median

## [1] 7.647628e-06

Number of selected edges with the median as threshold.

length(CLL_weights[CLL_weights >= CLL_meth_median])

## [1] 3630

Calculate the third quantile of the weights:

CLL_meth_q75 <- quantile(x = CLL_weights, 0.75)
CLL_meth_q75

##          75% 
## 1.159533e-05

Number of selected edges with the third quantile as threshold.

length(CLL_weights[CLL_weights >= CLL_meth_q75])

## [1] 1815

The following figure displays the log-transformed weight distribution with the two previous calculated metrics as markers.

hist(log(CLL_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "log10(weights)")
abline(v = log(CLL_meth_median, 10), col = "blue", lwd = 3)
text(log(CLL_meth_median, 10), 200, pos = 2, "Median", col = "blue", cex = 1)
abline(v = log(CLL_meth_q75, 10), col = "purple", lwd = 3)
text(log(CLL_meth_q75, 10), 230, pos = 4, "quantile 75%", col = "purple", cex = 1)

12.4.3.3 Topology network

To determine a range of thresholds to try, we check the weights.

summary(c(CLL_meth_W))

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 7.822e-07 4.287e-06 7.720e-06 9.893e-06 1.169e-05 2.782e-04

We define a vector of threshold range to try (at least 100 values).

thresholds <- seq(0, 0.00006, 0.0000005)
length(thresholds)

## [1] 121

Then, we calculate the Average Clustering Coefficient for each threshold.

CLL_ACC_meth <- do.call(rbind, lapply(thresholds, function(t, graph){
  graph_sub <- subgraph.edges(graph, E(graph)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(graph_sub), "thresholds" = t, "EN" = length(E(graph_sub)))
  return(df)
}, CLL_meth_net))

Calculated values are displayed in the following figures.

## ACC
plot(x = CLL_ACC_meth$thresholds, y = CLL_ACC_meth$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC calculation of the Fused network W", type = "o")
points(x = CLL_ACC_meth$thresholds[1], y = CLL_ACC_meth$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = CLL_ACC_meth$thresholds[41], y = CLL_ACC_meth$ACC[41], col = "pink", pch = 16, cex = 1.2)
points(x = CLL_ACC_meth$thresholds[42], y = CLL_ACC_meth$ACC[42], col = "purple", pch = 16, cex = 1.2)
text(CLL_ACC_meth$thresholds[42], 0.55, pos = 4, paste0("Threshold = ",  CLL_ACC_meth$thresholds[42]), col = "purple")
text(CLL_ACC_meth$thresholds[42], 0.45, pos = 4, paste0("ACCmax = ",  CLL_ACC_meth$ACC[42]), col = "purple")
## EN
plot(x = CLL_ACC_meth$thresholds, y = CLL_ACC_meth$EN, xlab = "thresholds", ylab = "number of edges", main = "EN of the Fused network W", type = "o")
abline(v = CLL_ACC_meth$thresholds[42], col = "purple")
text(CLL_ACC_meth$thresholds[42], 5500, pos = 4, paste0("Threshold = ",  CLL_ACC_meth$thresholds[42]), col = "purple")
text(CLL_ACC_meth$thresholds[42], 5000, pos = 4, paste0("ACCmax = ",  CLL_ACC_meth$ACC[42]), col = "purple")
text(CLL_ACC_meth$thresholds[42], 4500, pos = 4, paste0("EN = ",  CLL_ACC_meth[42, "EN"]), col = "purple")

Figure 12.4: Left: Average Clustering Coeeficient (ACC) values for each threshold - Right: Number of edges on network for each threshold

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.

If we selected the purple local maxima, we will have 164 edges. It could be not enough edges. Let’s see during the visualization.

CLL_ACC_meth$thresholds[42]

## [1] 2.05e-05

Network visualizations are available in the CLL_cytoscape.cys file.

12.4.4 mRNA data

CLL_mrna_net <- graph_from_adjacency_matrix(CLL_mrna_W, weighted = TRUE, mode = "upper", diag = FALSE)
write.table(as_data_frame(CLL_mrna_net), "../02_Results/02_CLL/CLL_mrna_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

12.4.4.1 Arbitrary threshold

CLL_weights <- edge.attributes(CLL_mrna_net)$weight
hist(CLL_weights, nclass = 100, main = "Fused network weight distribution", xlab = "weights")
hist(log(CLL_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "weights")
abline(v = log(5.011872e-06, 10), col = "cyan", lwd = 3)

The log-transformed weight distribution shows a kind of binomial distribution. We probably would like to cut between the two peaks and choose the corresponding weight: 5.011872e-06. With this threshold, we select 4013 connections.

12.4.4.2 Mean and third quantile

We calculate the median of the weights.

CLL_mrna_median <- median(x = CLL_weights)
CLL_mrna_median

## [1] 5.603111e-06

Number of selected edges with the median as threshold.

length(CLL_weights[CLL_weights >= CLL_mrna_median])

## [1] 3630

Calculate the third quantile of the weights:

CLL_mrna_q75 <- quantile(x = CLL_weights, 0.75)
CLL_mrna_q75

##          75% 
## 8.912064e-06

Number of selected edges with the third quantile as threshold.

length(CLL_weights[CLL_weights >= CLL_mrna_q75])

## [1] 1815

The following figure displays the log-transformed weight distribution with the two previous calculated metrics as markers.

hist(log(CLL_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "log10(weights)")
abline(v = log(CLL_mrna_median, 10), col = "blue", lwd = 3)
text(log(CLL_mrna_median, 10), 200, pos = 2, "Median", col = "blue", cex = 1)
abline(v = log(CLL_mrna_q75, 10), col = "purple", lwd = 3)
text(log(CLL_mrna_q75, 10), 190, pos = 4, "quantile 75%", col = "purple", cex = 1)

12.4.4.3 Topology network

To determine a range of thresholds to try, we check the weights.

summary(c(CLL_mrna_W))

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 3.123e-07 3.351e-06 5.647e-06 8.180e-06 9.049e-06 2.305e-04

We define a vector of threshold range to try (at least 100 values).

thresholds <- seq(0, 0.00004, 0.0000004)
length(thresholds)

## [1] 101

Then, we calculate the Average Clustering Coefficient for each threshold.

CLL_ACC_mrna <- do.call(rbind, lapply(thresholds, function(t, graph){
  graph_sub <- subgraph.edges(graph, E(graph)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(graph_sub), "thresholds" = t, "EN" = length(E(graph_sub)))
  return(df)
}, CLL_mrna_net))

Calculated values are displayed in the following figures.

## ACC
plot(x = CLL_ACC_mrna$thresholds, y = CLL_ACC_mrna$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC calculation of the Fused network W", type = "o")
points(x = CLL_ACC_mrna$thresholds[1], y = CLL_ACC_mrna$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = CLL_ACC_mrna$thresholds[46], y = CLL_ACC_mrna$ACC[46], col = "pink", pch = 16, cex = 1.2)
points(x = CLL_ACC_mrna$thresholds[48], y = CLL_ACC_mrna$ACC[48], col = "purple", pch = 16, cex = 1.2)
text(CLL_ACC_mrna$thresholds[48], 0.55, pos = 4, paste0("Threshold = ",  CLL_ACC_mrna$thresholds[48]), col = "purple")
text(CLL_ACC_mrna$thresholds[48], 0.45, pos = 4, paste0("ACCmax = ",  CLL_ACC_mrna$ACC[48]), col = "purple")
## EN
plot(x = CLL_ACC_mrna$thresholds, y = CLL_ACC_mrna$EN, xlab = "thresholds", ylab = "number of edges", main = "EN of the Fused network W", type = "o")
abline(v = CLL_ACC_mrna$thresholds[48], col = "purple")
text(CLL_ACC_mrna$thresholds[48], 5500, pos = 4, paste0("Threshold = ",  CLL_ACC_mrna$thresholds[48]), col = "purple")
text(CLL_ACC_mrna$thresholds[48], 5000, pos = 4, paste0("ACCmax = ",  CLL_ACC_mrna$ACC[48]), col = "purple")
text(CLL_ACC_mrna$thresholds[48], 4500, pos = 4, paste0("EN = ",  CLL_ACC_mrna[48, "EN"]), col = "purple")

Figure 12.5: Left: Average Clustering Coeeficient (ACC) values for each threshold - Right: Number of edges on network for each threshold

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.

If we selected the purple local maxima, we will have 229 edges. It could be not enough edges. Let’s see during the visualization.

CLL_ACC_mrna$thresholds[48]

## [1] 1.88e-05

Network visualizations are available in the CLL_cytoscape.cys file.

12.5 Downstream analysis

12.5.1 Clustering

We decided to perform a clustering analysis with two and three clusters.

12.5.1.1 With 2 clusters

C <- 2
group <- data.frame(Groups = spectralClustering(CLL_W, C)) 
row.names(group) <- colnames(CLL_W) 
CLL_dataGroups2 <- merge(CLL_metadata, group, by = 0)

12.5.1.2 With 3 clusters

C <- 3
group <- data.frame(Groups = spectralClustering(CLL_W, C)) 
row.names(group) <- colnames(CLL_W) 
CLL_dataGroups3 <- merge(CLL_metadata, group, by = 0)

12.5.1.3 Save results

Then, results are save into the same file.

clusters <- merge(x = CLL_dataGroups2, y = CLL_dataGroups3[c(1,7)], by = "Row.names", suffixes = c("_2clusters", "_3clusters"))
write.table(clusters, "../02_Results/02_CLL/CLL_clusters.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

12.5.2 Visualization with Cytoscape

These are two examples of network visualization for the CLL dataset.

Node color represents
- IGVH status (left network)
- clustering results (right network), we selected two clusters.
Node shape represents trisomy12 status.
Node labels are sample names.
Edge color represents data type contribution for each edge.

IGHV status is driving the clustering (right network). Inside this two groups, we can see a subnetwork that contains sample with trisomy12.

With this network, we can predicted the possible IGVH status of samples without this information.

13 Tomato plant dataset

The omic tomato plant dataset contains two omics data types:

transcript data
protein data

Data files are available in the summer school’s GitHub repository.

13.1 Input data

13.1.1 Load dataset

13.1.1.1 Transcript data

First, we load the transcript data that are in the mrna.tsv file. This file contains header (head = TRUE) and the first column contains row names (row.names = 1). Below, the first columns and rows are displayed.

tomato_mrna <- read.table("../00_Data/Tomato/mrna.tsv", head = TRUE, row.names = 1)
tomato_mrna[c(1:5), c(1:5)]

##                     s_1_1  s_1_2  s_1_3  s_2_1  s_2_2
## Solyc00g005050.2.1 -0.542 -0.431 -0.519 -0.128 -0.055
## Solyc00g006800.2.1 -0.243 -0.234 -0.165  0.060  0.177
## Solyc00g007270.2.1 -0.641 -0.738 -0.761 -0.331 -0.128
## Solyc00g009020.2.1  0.346  0.607  0.155 -0.131 -0.081
## Solyc00g011890.2.1 -0.318 -0.419 -0.510 -0.484 -0.479

Transcript data dimensions are nrow(tomato_mrna) rows and ncol(tomato_mrna):

dim(tomato_mrna)

## [1] 2375   27

Samples (2375) are in columns and transcripts (2375) are in rows. We need to transpose this matrix.

tomato_mrna_t <- t(tomato_mrna)

The transposed matrix dimensions are nrow(tomato_mrna_t) rows and ncol(tomato_mrna_t). Samples are in columns and transcripts are in rows.

dim(tomato_mrna_t)

## [1]   27 2375

Below, the first five rows and columns are displayed:

tomato_mrna_t[c(1:5), c(1:5)]

##       Solyc00g005050.2.1 Solyc00g006800.2.1 Solyc00g007270.2.1
## s_1_1             -0.542             -0.243             -0.641
## s_1_2             -0.431             -0.234             -0.738
## s_1_3             -0.519             -0.165             -0.761
## s_2_1             -0.128              0.060             -0.331
## s_2_2             -0.055              0.177             -0.128
##       Solyc00g009020.2.1 Solyc00g011890.2.1
## s_1_1              0.346             -0.318
## s_1_2              0.607             -0.419
## s_1_3              0.155             -0.510
## s_2_1             -0.131             -0.484
## s_2_2             -0.081             -0.479

13.1.1.2 Protein data

Then, we load the protein data, available in the file prots.tsv. The file contains column heads (head = TRUE) and the first column contains row names (row.names = 1). Below, the first five columns and rows are displayed.

tomato_prot <- read.table("../00_Data/Tomato/prots.tsv", head = TRUE, row.names = 1)
tomato_prot[c(1:5), c(1:5)]

##                     s_1_1  s_1_2  s_1_3  s_2_1  s_2_2
## Solyc00g005050.2.1 -0.025  0.695  0.270 -0.031  0.146
## Solyc00g006800.2.1 -0.391 -0.457 -0.003  0.217 -0.004
## Solyc00g007270.2.1 -1.469 -0.775 -1.311 -0.185 -0.061
## Solyc00g009020.2.1  0.562  0.493  0.895  0.525  0.471
## Solyc00g011890.2.1  0.063 -0.078  0.119 -0.346 -0.339

Protein data have 27 columns:

ncol(tomato_prot)

## [1] 27

Protein data have 2375 rows:

nrow(tomato_prot)

## [1] 2375

Rows are proteins and columns are samples. To continue the analysis, data need to have the samples in rows and features in columns. So, we transpose the protein data.

tomato_prot_t <- t(tomato_prot)

Now, protein data are in the right shape, as you can see below:

tomato_prot_t[c(1:5), c(1:5)]

##       Solyc00g005050.2.1 Solyc00g006800.2.1 Solyc00g007270.2.1
## s_1_1             -0.025             -0.391             -1.469
## s_1_2              0.695             -0.457             -0.775
## s_1_3              0.270             -0.003             -1.311
## s_2_1             -0.031              0.217             -0.185
## s_2_2              0.146             -0.004             -0.061
##       Solyc00g009020.2.1 Solyc00g011890.2.1
## s_1_1              0.562              0.063
## s_1_2              0.493             -0.078
## s_1_3              0.895              0.119
## s_2_1              0.525             -0.346
## s_2_2              0.471             -0.339

13.1.2 Load metadata

Finally, we load the metadata that contain information about samples. Metadata are stored in the samples_metadata.csv file. This file is semicolon-separated (sep = ";") and contains column heads (head = TRUE). The first column is also row names (row.names = 1).

tomato_metadata <- read.table("../00_Data/Tomato/samples_metadata.csv", head = TRUE, row.names = 1, sep = ";")

The metadata file contains information about:

dpa: days post anthesis (after the flower opening)
growth_stage: growth stage of the tomato plant

names(tomato_metadata)

## [1] "dpa"          "growth_stage"

There are 20 different dpa:

unique(tomato_metadata$dpa)

##  [1]  7  8 15 21 22 27 28 29 34 35 40 42 49 48 NA 50 51 54 53 52

There are three replicates per growth stage:

table(tomato_metadata$growth_stage)

## 
## GR1 GR2 GR3 GR4 GR5 GR6 GR7 GR8 GR9 
##   3   3   3   3   3   3   3   3   3

13.1.3 Missing data

Transcript data don’t contain missing value:

table(is.na(tomato_mrna_t))

## 
## FALSE 
## 64125

Protein data don’t contain neither missing value:

table(is.na(tomato_prot_t))

## 
## FALSE 
## 64125

We can go to the following steps.

13.1.4 Scaling

We assume that data habe been already prepared and normalized.

13.1.4.1 Transcript data

Transcript data are scaled: each column will scaled to have the mean equals to zero and the standard deviation equals to one.

tomato_mrna_scaled <- standardNormalization(tomato_mrna_t)

Below, we show the data distribution before and after scaling. We expected to have a normal distribution of the data after scaling.

hist(tomato_mrna_t, nclass = 100, main = "Tomato fruit - Transcript data - Prepared data", xlab = "values")
hist(tara_phy_scaled, nclass = 100, main = "Tomato fruit - Transcript data - Scaled data", xlab = "values")

Data seem to be already scaled. So we will use the transposed data tomato_mrna_t for the following analysis.

13.1.4.2 Protein data

Protein data are scaled.

tomato_prot_scaled <- standardNormalization(tomato_prot_t)

Below, we show the data distribution before and after scaling. We expected to have a normal distribution of the data after scaling.

hist(tomato_prot_t, nclass = 100, main = "Tomato fruit - Protein data - Prepared data", xlab = "values")
hist(tomato_prot_scaled, nclass = 100, main = "Tomato fruit - Protein data - Scaled data", xlab = "values")

These data seem also already scaled. So we will use the transposed data tomato_prot_t for the following analysis.

13.2 Similarity network

In this part, we create the similarity network for each data type.

13.2.1 Distance calculation

First, we calculate the Euclidean distance between each sample for each data type.

tomato_mrna_dist <- dist2(tomato_mrna_t, tomato_mrna_t)
tomato_prot_dist <- dist2(tomato_prot_t, tomato_prot_t)

The created distance matrix dimensions are 27 rows and 27 columns. We calculated pairwise distance, so the matrix has samples in rows and in columns.

dim(tomato_mrna_dist)

## [1] 27 27

The diagonal of the distance matrix contains the distance between sample and itself. There is distance between a sample and itself, so the distance is equal (or very close) to zero.

tomato_mrna_dist[c(1:5), c(1:5)]

##              s_1_1        s_1_2        s_1_3        s_2_1        s_2_2
## s_1_1 9.094947e-13 1.326255e+02 8.693254e+01 2.401490e+02 3.900077e+02
## s_1_2 1.326255e+02 6.821210e-13 2.801969e+02 3.532615e+02 4.989396e+02
## s_1_3 8.693254e+01 2.801969e+02 2.046363e-12 2.096528e+02 3.614704e+02
## s_2_1 2.401490e+02 3.532615e+02 2.096528e+02 1.023182e-12 4.759286e+01
## s_2_2 3.900077e+02 4.989396e+02 3.614704e+02 4.759286e+01 1.136868e-13

High distance values mean that samples are not similar. And small distance values mean that samples are similar.

13.2.2 Similarity calculation

The distance matrix is then transformed into similarity matrix for each data type. We set two parameters:

K = 20: number of nearest neighbors
signma = 0.5: hyperparameter

K <- 20
sigma <- 0.5

The affinityMatrix() function transforms the distance into similarity according the distance with the nearest neighbors.

tomato_mrna_W <- affinityMatrix(tomato_mrna_dist, K = K, sigma = sigma)
tomato_prot_W <- affinityMatrix(tomato_prot_dist, K = K, sigma = sigma)

The following figures are the heatmap of the similarity matrix (W) of each data type. The left heatmap are the mrna data and the right heatmap are the protein data. Samples are clustered using hierarchical clustering. For a better visualization, we log-transform similarities.

pheatmap(log(tomato_mrna_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = tomato_metadata, main = "Transcript data - log10-transformed similarity values")
pheatmap(log(tomato_prot_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = tomato_metadata, main = "Protein data - log10-transformed similarity values")

Red color means a high similarity value between two samples whereas blue color means a small similarity value between two samples.

Heatmaps are different between the transcript and the protein data. Each data type carries different kind of sample information. The clustering shows that one group is clearly retrieve in both data type (samples in last dpa). Protein data seem to define better the development cycle of the tomato fruit.

13.3 Fusion

13.3.1 Create the fused similarity matrix

We create the fused similarity matrix using these three parameters:

the list of mrna and protein similarity matrices
K = 20: number of nearest neighbors
t = 10: number of iterations

tomato_W <- SNF(list(tomato_mrna_W, tomato_prot_W), K = 20, t = 10)

The dimension of the fused network are 27 rows and 27 columns, such as the previous similarity matrices. The fused similarity matrix contains similarities between samples, we can also called them weights.

The fused similarity network contains 729 weights.

length(tomato_W)

## [1] 729

The fused similarity network doesn’t contain zero:

table(tomato_W == 0)

## 
## FALSE 
##   729

The following figure is the heatmap of the fused similarity matrix. Samples are automatically clustered with a hierarchical clustering. Weights are log-transformed for a better visualization.

pheatmap(log(tomato_W, 10), show_rownames = FALSE, show_colnames = FALSE, annotation = tomato_metadata, main = "Fused similarity matrix - log10-transformed similarity values")

Read color means a high similarity between samples. Blue color means a small similarity between samples.

This heatmap seems to be a perfect mix between the two previous individual heatmaps. We still see two main groups: one with the last dpa and one other big with the other development stages. But, in this big group, now stages seem to be well grouped.

13.3.2 Visualize the fused similarity network

We create a fused similarity network from the fused similarity matrix. Self loops are remove (diag = FALSE) and only the upper values of the matrix are taken (mode = "upper", avoid duplicate information).

tomato_W_net <- graph_from_adjacency_matrix(tomato_W, diag = FALSE, mode = "upper", weighted = TRUE)

Then, the fused similarity network is saved into a the Tomato_W_edgeList.txt file:

write.table(as_data_frame(tomato_W_net), "../02_Results/04_Tomato/Tomato_W_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

This files is loaded into Cytoscape. The Figure ?? shows the fused similarity network of the tomato plant dataset.

Figure 13.1: First Cytoscape visualization of the fused similarity network of Tomato fruit dataset.

According Cytoscape, the fused network contains 27 samples (nodes) and 351 connections (edges) between samples. The edge number is smaller in Cytoscape because we removed the self loop and took only half of the similarity matrix.

13.4 Threshold selection

So in this section, we will choose a threshold to keep the strongest connections.

13.4.1 Fused network

13.4.1.1 Arbitrary threshold

tomato_weights <- edge.attributes(tomato_W_net)$weight
hist(tomato_weights, nclass = 100, main = "Fused similarity network weight distribution", xlab = "weights")
hist(log(tomato_weights, 10), nclass = 100, main = "Fused similarity network weight distribution", xlab = "weights")

It’s not obvious how to choose the threshold with the weight distribution. Let’s see other methods.

13.4.1.2 Mean and third quantile

Calculate the median:

tomato_W_median <- median(x = tomato_weights)
tomato_W_median

## [1] 0.01292061

Number of selected edges with the median as threshold:

length(tomato_weights[tomato_weights >=  tomato_W_median])

## [1] 176

Calculate the third quantile:

tomato_W_q75 <- quantile(x = tomato_weights, 0.75)
tomato_W_q75

##       75% 
## 0.0324337

Number of selected edges with the third quantile as threshold:

length(tomato_weights[tomato_weights >=  tomato_W_q75])

## [1] 88

The following figures show where are these two threshold in the weight distribution.

hist(log(tomato_weights, 10), nclass = 100, main = "Fused network weight distribution", xlab = "log10(weights)")
abline(v = log(tomato_W_median, 10), col = "blue", lwd = 3)
text(log(tomato_W_median, 10), 10, pos = 2, "Median", col = "blue", cex = 1)
abline(v = log(tomato_W_q75, 10), col = "purple", lwd = 3)
text(log(tomato_W_q75, 10), 12, pos = 4, "quantile 75%", col = "purple", cex = 1)

13.4.1.3 Topology network

To determine the range of the threshold, we check the weights:

summary(tomato_weights)

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## 0.001066 0.003292 0.012921 0.019231 0.032434 0.095629

We define the threshold range to try:

thresholds <- seq(0, 0.095629, 0.0005)
length(thresholds)

## [1] 192

Then, we calculate the Average Clustering Coefficient for each threshold.

tomato_ACC_W <- do.call(rbind, lapply(thresholds, function(t, net){
  net_sub <- subgraph.edges(net, E(net)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(net_sub), "thresholds" = t, "EN" = length(E(net_sub)))
  return(df)
}, tomato_W_net))

Calculated values are displayed in the following figures:

plot(x = tomato_ACC_W$thresholds, y = tomato_ACC_W$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC calculation of the Fused network W", type = "o")
points(x = tomato_ACC_W$thresholds[1], y = tomato_ACC_W$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = tomato_ACC_W$thresholds[73], y = tomato_ACC_W$ACC[73], col = "pink", pch = 16, cex = 1.2)
points(x = tomato_ACC_W$thresholds[80], y = tomato_ACC_W$ACC[80], col = "purple", pch = 16, cex = 1.2)
points(x = tomato_ACC_W$thresholds[71], y = tomato_ACC_W$ACC[71], col = "cyan", pch = 16, cex = 1.2)
abline(v = tomato_ACC_W$thresholds[80], col = "purple")
text(tomato_ACC_W$thresholds[80], 0.5, pos = 4, paste0("Threshold = ",  tomato_ACC_W$thresholds[80]), col = "purple")
text(tomato_ACC_W$thresholds[80], 0.4, pos = 4, paste0("ACCmax = ",  tomato_ACC_W$ACC[80]), col = "purple")
plot(x = tomato_ACC_W$thresholds, y = tomato_ACC_W$EN, xlab = "thresholds", ylab = "number of edges", main = "EN of the Fused network W", type = "o")
abline(v = tomato_ACC_W$thresholds[80], col = "purple")
text(tomato_ACC_W$thresholds[80], 350, pos = 4, paste0("Threshold = ",  tomato_ACC_W$thresholds[80]), col = "purple")
text(tomato_ACC_W$thresholds[80], 300, pos = 4, paste0("ACCmax = ",  tomato_ACC_W$ACC[80]), col = "purple")
text(tomato_ACC_W$thresholds[80], 250, pos = 4, paste0("EN = ",  tomato_ACC_W[80, "EN"]), col = "purple")

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.
The blue dot value is another local maxima.

If we selected the purple local maxima, we will have 52 edges. It could be not enough edges.

tomato_ACC_W$thresholds[80]

## [1] 0.0395

We can try with another local maxima.

tomato_ACC_W$thresholds[71]

## [1] 0.035

tomato_ACC_W$EN[71]

## [1] 72

It could be interesting to try another threshold more.

tomato_ACC_W$thresholds[18]

## [1] 0.0085

tomato_ACC_W$EN[18]

## [1] 203

The following figures are filtered network using 0.035 (left) and 0.013 (right) as thresholds.

We think that the right network doesn’t have enough edges. We loose to much information between samples. We will probably use the network on the left for the following visualization.

Network visualizations are available in the Tomato_cytoscape.cys file.

13.4.2 Transcript data

13.4.2.1 Arbitrary threshold

tomato_mrna_net <- graph_from_adjacency_matrix(tomato_mrna_W, diag = FALSE, mode = "upper", weighted = TRUE)
write.table(as_data_frame(tomato_mrna_net), "../02_Results/04_Tomato/Tomato_mrna_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

It’s not obvious how to choose the threshold with the weight distribution. Let’s see other methods.

tomato_weights <- edge.attributes(tomato_mrna_net)$weight
hist(tomato_weights, nclass = 100, main = "Transcript weight distribution", xlab = "weights")
hist(log(tomato_weights, 10), nclass = 100, main = "Transcript weight distribution", xlab = "weights")

13.4.2.2 Mean and third quantile

Calculate the median:

tomato_mrna_median <- median(x = tomato_weights)
tomato_mrna_median

## [1] 6.307718e-05

Number of selected edges with the median as threshold:

length(tomato_weights[tomato_weights >=  tomato_mrna_median])

## [1] 176

Calculate the third quantile:

tomato_mrna_q75 <- quantile(x = tomato_weights, 0.75)
tomato_mrna_q75

##          75% 
## 0.0006490944

Number of selected edges with the third quantile as threshold:

length(tomato_weights[tomato_weights >=  tomato_mrna_q75])

## [1] 88

The following figures show where are these two threshold in the weight distribution.

hist(log(tomato_weights, 10), nclass = 100, main = "Transcript weight distribution", xlab = "log10(weights)")
abline(v = log(tomato_mrna_median, 10), col = "blue", lwd = 3)
text(log(tomato_mrna_median, 10), 10, pos = 2, "Median", col = "blue", cex = 1)
abline(v = log(tomato_mrna_q75, 10), col = "purple", lwd = 3)
text(log(tomato_mrna_q75, 10), 12, pos = 2, "quantile 75%", col = "purple", cex = 1)

13.4.2.3 Topology network

To determine the range of the threshold, we check the weights:

summary(tomato_weights)

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 1.571e-06 8.678e-06 6.308e-05 3.722e-04 6.491e-04 2.522e-03

We define the threshold range to try:

thresholds <- seq(0, 2.522e-03, 0.00002)
length(thresholds)

## [1] 127

Then, we calculate the Average Clustering Coefficient for each threshold.

tomato_ACC_mrna <- do.call(rbind, lapply(thresholds, function(t, net){
  net_sub <- subgraph.edges(net, E(net)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(net_sub), "thresholds" = t, "EN" = length(E(net_sub)))
  return(df)
}, tomato_mrna_net))

Calculated values are displayed in the following figures:

plot(x = tomato_ACC_mrna$thresholds, y = tomato_ACC_mrna$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC calculation of the Transcript W", type = "o")
points(x = tomato_ACC_mrna$thresholds[1], y = tomato_ACC_mrna$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = tomato_ACC_mrna$thresholds[36], y = tomato_ACC_mrna$ACC[36], col = "pink", pch = 16, cex = 1.2)
points(x = tomato_ACC_mrna$thresholds[37], y = tomato_ACC_mrna$ACC[37], col = "purple", pch = 16, cex = 1.2)
abline(v = tomato_ACC_mrna$thresholds[37], col = "purple")
text(tomato_ACC_mrna$thresholds[37], 0.5, pos = 2, paste0("Threshold = ",  tomato_ACC_mrna$thresholds[37]), col = "purple")
text(tomato_ACC_mrna$thresholds[37], 0.4, pos = 2, paste0("ACCmax = ",  tomato_ACC_mrna$ACC[37]), col = "purple")
plot(x = tomato_ACC_mrna$thresholds, y = tomato_ACC_mrna$EN, xlab = "thresholds", ylab = "number of edges", main = "EN of the Transcript W", type = "o")
abline(v = tomato_ACC_mrna$thresholds[37], col = "purple")
text(tomato_ACC_mrna$thresholds[37], 350, pos = 4, paste0("Threshold = ",  tomato_ACC_mrna$thresholds[37]), col = "purple")
text(tomato_ACC_mrna$thresholds[37], 300, pos = 4, paste0("ACCmax = ",  tomato_ACC_mrna$ACC[37]), col = "purple")
text(tomato_ACC_mrna$thresholds[37], 250, pos = 4, paste0("EN = ",  tomato_ACC_mrna[37, "EN"]), col = "purple")

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.

The local maxima threshold is:

tomato_ACC_mrna$thresholds[37]

## [1] 0.00072

And the number of selected egdes are:

tomato_ACC_mrna$EN[37]

## [1] 77

Network visualizations are available in the Tomato_cytoscape.cys file.

13.4.3 Protein data

13.4.3.1 Arbitrary threshold

tomato_prot_net <- graph_from_adjacency_matrix(tomato_prot_W, diag = FALSE, mode = "upper", weighted = TRUE)
write.table(as_data_frame(tomato_prot_net), "../02_Results/04_Tomato/Tomato_prot_edgeList.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

It’s not obvious how to choose the threshold with the weight distribution. Let’s see other methods.

tomato_weights <- edge.attributes(tomato_prot_net)$weight
hist(tomato_weights, nclass = 100, main = "Protein weight distribution", xlab = "weights")
hist(log(tomato_weights, 10), nclass = 100, main = "Protein weight distribution", xlab = "weights")

13.4.3.2 Mean and third quantile

Calculate the median:

tomato_prot_median <- median(x = tomato_weights)
tomato_prot_median

## [1] 7.254911e-05

Number of selected edges with the median as threshold:

length(tomato_weights[tomato_weights >=  tomato_prot_median])

## [1] 176

Calculate the third quantile:

tomato_prot_q75 <- quantile(x = tomato_weights, 0.75)
tomato_prot_q75

##          75% 
## 0.0004788178

Number of selected edges with the third quantile as threshold:

length(tomato_weights[tomato_weights >=  tomato_prot_q75])

## [1] 88

The following figures show where are these two threshold in the weight distribution.

hist(log(tomato_weights, 10), nclass = 100, main = "Protein weight distribution", xlab = "log10(weights)")
abline(v = log(tomato_mrna_median, 10), col = "blue", lwd = 3)
text(log(tomato_mrna_median, 10), 12, pos = 2, "Median", col = "blue", cex = 1)
abline(v = log(tomato_mrna_q75, 10), col = "purple", lwd = 3)
text(log(tomato_mrna_q75, 10), 12, pos = 2, "quantile 75%", col = "purple", cex = 1)

13.4.3.3 Topology network

To determine the range of the threshold, we check the weights:

summary(tomato_weights)

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 2.412e-06 1.379e-05 7.255e-05 3.042e-04 4.788e-04 2.169e-03

We define the threshold range to try:

thresholds <- seq(0, 0.001, 0.00001)
length(thresholds)

## [1] 101

Then, we calculate the Average Clustering Coefficient for each threshold.

tomato_ACC_prot <- do.call(rbind, lapply(thresholds, function(t, net){
  net_sub <- subgraph.edges(net, E(net)[weight >= t])
  df <- data.frame("ACC" = ACCCalculation(net_sub), "thresholds" = t, "EN" = length(E(net_sub)))
  return(df)
}, tomato_prot_net))

Calculated values are displayed in the following figures:

plot(x = tomato_ACC_prot$thresholds, y = tomato_ACC_prot$ACC, xlab = "thresholds", ylab = "ACC", main = "ACC calculation of the protein W", type = "o")
points(x = tomato_ACC_prot$thresholds[1], y = tomato_ACC_prot$ACC[1], col = "red", pch = 16, cex = 1.2)
points(x = tomato_ACC_prot$thresholds[7], y = tomato_ACC_prot$ACC[7], col = "pink", pch = 16, cex = 1.2)
points(x = tomato_ACC_prot$thresholds[9], y = tomato_ACC_prot$ACC[9], col = "purple", pch = 16, cex = 1.2)
abline(v = tomato_ACC_prot$thresholds[9], col = "purple")
text(tomato_ACC_prot$thresholds[9], 0.8, pos = 4, paste0("Threshold = ",  tomato_ACC_prot$thresholds[9]), col = "purple")
text(tomato_ACC_prot$thresholds[9], 0.7, pos = 4, paste0("ACCmax = ",  tomato_ACC_prot$ACC[9]), col = "purple")
plot(x = tomato_ACC_prot$thresholds, y = tomato_ACC_prot$EN, xlab = "thresholds", ylab = "number of edges", main = "EN of the protein W", type = "o")
abline(v = tomato_ACC_prot$thresholds[9], col = "purple")
text(tomato_ACC_prot$thresholds[9], 350, pos = 4, paste0("Threshold = ",  tomato_ACC_prot$thresholds[9]), col = "purple")
text(tomato_ACC_prot$thresholds[9], 300, pos = 4, paste0("ACCmax = ",  tomato_ACC_prot$ACC[9]), col = "purple")
text(tomato_ACC_prot$thresholds[9], 250, pos = 4, paste0("EN = ",  tomato_ACC_prot[9, "EN"]), col = "purple")

The red dot value corresponds to the fully connected network.
The pink dot value is the smallest value before the local maxima.
The purple dot value is the local maxima. The one we are interested in.

The local maxima threshold is:

tomato_ACC_prot$thresholds[9]

## [1] 8e-05

And the number of selected egdes are:

tomato_ACC_prot$EN[9]

## [1] 170

Network visualizations are available in the Tomato_cytoscape.cys file.

13.5 Downstream analysis

13.5.1 Clustering

In the Belouah et al. paper, they define three development stages. According that, we will run a clustering with three clusters.

C <- 3
group <- data.frame(Groups = spectralClustering(tomato_W, C)) 
row.names(group) <- colnames(tomato_W)
dataGroups <- merge(tomato_metadata, group, by = 0) 
head(dataGroups)

##   Row.names dpa growth_stage Groups
## 1     s_1_1   7          GR1      3
## 2     s_1_2   8          GR1      3
## 3     s_1_3   8          GR1      3
## 4     s_2_1  15          GR2      3
## 5     s_2_2  15          GR2      3
## 6     s_2_3  15          GR2      3

Then, we save the results into a result file:

write.table(dataGroups, "../02_Results/04_Tomato/Tomato_3clusters.txt", quote = FALSE, col.names = TRUE, row.names = FALSE, sep = "\t")

13.5.2 Visualization with Cytoscape

These are two examples of network visualization for the tomato fruit dataset.

Figure 13.2: Left network: edge weights > 0.013. Right network: edge weights > 0.032.

Node color represents the clustering results. Here, we selected three clusters.
Node label are the growth stat of the tomato fruit.
Edge color represents the data type contribution for each edge.

Overall, we see two groups of nodes: one with the late growth stages (GR7, GR8 and GR9) and one other with the early growth stages (GR1-GR6). This two groups seem to be very different because there are few connections between them. We already saw these two groups with the individual heatmaps.

In the Belouah et al. paper, the three last growth state correspond to the ripening stage (appearance of fruit color). With the clustering, we also detect the three stage level that they describe: early, mid and late stages of fruit development.

With the network visualization, we can see a kind of kinetic from the early stage (GR1 only connected to GR2) to the mid stage. We can also see that the protein data type carries the most information in this network, inside the groups.

Similarity Network Fusion (SNF)

Morgane Térézol - Galadriel Brière

septembre 01, 2023

1 Libraries and environment

1.1 Load environment

1.2 Load libraries

1.3 Visualization using Cytoscape

2 General principle of the SNF method

3 Choose your datasets

3.1 Metagenomic dataset from Tara Ocean project

3.2 Breast cancer dataset from The Cancer Genome Atlas

3.3 Chronic Lymphocytic Leukaemia (CLL) dataset

3.4 Tomato plant dataset

4 Input data

4.1 In summary

4.2 Read and prepare the data

4.2.1 Load dataset

4.2.1.1 Load from file

4.2.1.2 Load from package

4.2.1.3 Load from website

4.2.2 Load metadata

4.2.3 Practice

4.3 Missing data

4.4 Normalization and scaling

4.4.1 Normalization

4.4.2 Scaling

5 Similarity network

5.1 In summary

5.2 Distance calculation

5.3 Similarity calculation

6 Fusion

6.1 In summary

6.2 Create the fused similarity matrix

6.3 Visualize the fused similarity network

6.3.1 Create the fused similarity network

6.3.2 Visualize using Cytoscape

6.3.2.1 Import files

6.3.2.2 Network style

6.3.2.3 Network layout

6.3.3 Practice

7 Threshold selection

7.1 In summary

7.2 Select an arbitrary threshold

7.3 Select the threshold using quantiles

7.4 Threshold based on network topology

7.4.1 Functions

7.4.2 Range of thresholds

7.4.3 Calculate the ACC values

7.5 Visualize this threshold in Cytoscape

7.5.1 Filter column

7.5.2 Create a new network visualization

7.6 Practice

8 Downstream analysis

8.1 Clustering

8.2 Visualization on Cytoscape

9 Go further

9.1 Normalization and distance calculation

9.2 Data type contribution

9.2.1 Data type contribution: with 2 data types

9.2.2 Data type contribution: with 3 data types

10 Tara Ocean dataset

10.1 Input data

10.1.1 Load dataset

10.1.1.1 Orthologous genes (nog) data

10.1.1.2 Phylogentic profil (phy) data

10.1.2 Load metadata

10.1.3 Missing data

10.1.4 Scaling

10.1.4.1 nog data

10.1.4.2 phy data

10.2 Similarity network

10.2.1 Distance calculation

10.2.2 Similarity calculation

10.3 Fusion

10.3.1 Create the fused similarity matrix

10.3.2 Visualized the fused similarity network

10.4 Threshold selection

10.4.1 Fused similarity network

10.4.1.1 Arbitrary threshold

10.4.1.2 Mean and third quantile